Algebra 2 Quiz: Comparing Functions Represented In Different Ways
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Comparing Functions Represented In Different WaysQuestion 1 of 20

Function ff is described verbally as: "An exponential function with initial value 55 and growth factor 1.51.5." Function gg is given algebraically by g(x)=4⋅2xg(x)=4\cdot 2^x.

Which function has the larger yy-intercept?

Function gg has the larger yy-intercept (44 vs 55).
Function ff has the larger yy-intercept (55 vs 44).
Function gg has the larger yy-intercept (88 vs 55).
They have equal yy-intercepts (both 55).
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Algebra 2 Quiz

Algebra 2 Quiz: Comparing Functions Represented In Different Ways

Practice Comparing Functions Represented In Different Ways in Algebra 2 with focused quiz questions that help you check what you know, review explanations, and build confidence with test-style prompts.

What this quiz covers

This quiz focuses on Comparing Functions Represented In Different Ways, giving you a quick way to practice the rules, question types, and explanations that matter most for Algebra 2.

How to use this quiz

Try each quiz question before looking at the correct answer. Use the explanations to review missed ideas, then come back to similar questions until the pattern feels familiar.

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Question 1

Function ff is described verbally as: "An exponential function with initial value 55 and growth factor 1.51.5." Function gg is given algebraically by g(x)=4⋅2xg(x)=4\cdot 2^x.

Which function has the larger yy-intercept?

  1. Function gg has the larger yy-intercept (44 vs 55).
  2. Function ff has the larger yy-intercept (55 vs 44). (correct answer)
  3. Function gg has the larger yy-intercept (88 vs 55).
  4. They have equal yy-intercepts (both 55).
Explanation: This question tests your ability to compare function properties when functions are presented in different ways—like one given as a formula and another as a graph—requiring you to extract the same feature from each representation and compare them. Functions can be represented in four main ways, and each representation makes certain features easy to see: formulas let you calculate any value precisely and see structure (like y = mx + b shows slope m immediately), graphs show shape and extrema visually (you can spot maximums and end behavior at a glance), tables provide exact input-output pairs (easy to read specific values), and verbal descriptions summarize key features in words. To compare functions in different representations, extract the desired property from each using the method that fits that representation, then compare the extracted values! For function f described verbally as exponential with initial value 5 (which is the y-intercept), it's directly given; for function g given by g(x) = 4 · 2^x, set x=0 to get 4 · 1 = 4, so comparing shows f's 5 is larger than g's 4. Choice A correctly extracts the y-intercept from both and compares accurately, showing function f has the larger y-intercept. A distractor like choice B might result from miscalculating g(0) as 8 (perhaps thinking 4·2 instead of 4⋅204·2^0=4)—always plug in x=0 carefully for the initial value. Property extraction by representation type: FROM FORMULA—y-intercept: set x = 0; maximum of quadratic: complete square or use vertex formula; slope: read from y = mx + b or calculate rise/run. FROM GRAPH—intercepts: see where crosses axes; maximum: find highest point and read coordinates; slope: pick two points, calculate rise/run. FROM TABLE—intercept: find row where x = 0 or y = 0; maximum: scan for largest y-value; slope (linear): calculate Δy/Δx between any two points. Use the method matching your representation! Quick comparison shortcuts: for y-intercepts, formulas are fastest (plug in x = 0), but graphs let you just read off where it crosses the y-axis. For maxima, graphs are easiest (visually find highest point), but formulas of quadratics give exact vertex. For growth rates, tables let you calculate differences (linear) or ratios (exponential) directly. Play to each representation's strengths—don't convert everything to formulas if the feature is obvious in the given form! Efficient comparison means using the representation smartly.

Question 2

Function ff is linear and given by f(x)=−4x+7f(x) = -4x + 7. Function gg is represented by the table.

\begin{center} \begin{tabular}{c|c} xx & g(x)g(x) \\hline 00 & 22 \ 11 & 55 \ 22 & 88 \ 33 & 1111 \ \end{tabular} \end{center}

Which function has the greater slope (rate of change)?

  1. Function ff
  2. Function gg (correct answer)
  3. They have equal slopes.
  4. Cannot be determined from the information given.
Explanation: This question tests your ability to compare function properties when functions are presented in different ways—like one given as a formula and another as a graph—requiring you to extract the same feature from each representation and compare them. Functions can be represented in four main ways, and each representation makes certain features easy to see: formulas let you calculate any value precisely and see structure (like y = mx + b shows slope m immediately), graphs show shape and extrema visually (you can spot maximums and end behavior at a glance), tables provide exact input-output pairs (easy to read specific values), and verbal descriptions summarize key features in words. To compare functions in different representations, extract the desired property from each using the method that fits that representation, then compare the extracted values! For function f given by f(x) = -4x + 7, the slope is immediately visible as the coefficient of x, which is -4. For function g given in the table, we calculate the slope using any two points: taking (0,2) and (1,5), slope = (5-2)/(1-0) = 3/1 = 3, or using (1,5) and (2,8), slope = (8-5)/(2-1) = 3/1 = 3. Choice B correctly identifies that g has the greater slope since 3 > -4 (remember that positive slopes are greater than negative slopes). A common error would be to compare absolute values and think |-4| > 3, but when comparing slopes, we must consider the sign—positive slopes indicate increasing functions while negative slopes indicate decreasing functions. Property extraction by representation type: FROM FORMULA—y-intercept: set x = 0; maximum of quadratic: complete square or use vertex formula; slope: read from y = mx + b or calculate rise/run. FROM GRAPH—intercepts: see where crosses axes; maximum: find highest point and read coordinates; slope: pick two points, calculate rise/run. FROM TABLE—intercept: find row where x = 0 or y = 0; maximum: scan for largest y-value; slope (linear): calculate Δy/Δx between any two points. Use the method matching your representation! Quick comparison shortcuts: for y-intercepts, formulas are fastest (plug in x = 0), but graphs let you just read off where it crosses the y-axis. For maxima, graphs are easiest (visually find highest point), but formulas of quadratics give exact vertex. For growth rates, tables let you calculate differences (linear) or ratios (exponential) directly. Play to each representation's strengths—don't convert everything to formulas if the feature is obvious in the given form! Efficient comparison means using the representation smartly.

Question 3

Function ff is given by f(x)=x2−6x+5f(x)=x^2-6x+5. Function gg is shown in the table:

x01234g(x)20−202\begin{array}{c|ccccc} x & 0 & 1 & 2 & 3 & 4\\ \hline g(x) & 2 & 0 & -2 & 0 & 2 \end{array}

Which statement correctly compares the minimum values of ff and gg?

  1. ff has a smaller minimum value than gg (−4-4 vs −2-2). (correct answer)
  2. gg has a smaller minimum value than ff (−2-2 vs −1-1).
  3. They have the same minimum value (−2-2).
  4. ff has a larger minimum value than gg (−4-4 vs −2-2).
Explanation: This question tests your ability to compare function properties when functions are presented in different ways—like one given as a formula and another as a table—requiring you to extract the same feature from each representation and compare them. Functions can be represented in four main ways, and each representation makes certain features easy to see: formulas let you calculate any value precisely and see structure (like y = mx + b shows slope m immediately), graphs show shape and extrema visually (you can spot maximums and end behavior at a glance), tables provide exact input-output pairs (easy to read specific values), and verbal descriptions summarize key features in words. To compare functions in different representations, extract the desired property from each using the method that fits that representation, then compare the extracted values! For function f, given as f(x) = x^2 - 6x + 5, find minimum by vertex formula x = -b/(2a) = 6/2 = 3, then f(3) = 9 - 18 + 5 = -4. For function g, scan table values (2, 0, -2, 0, 2) to find smallest is -2. Choice A correctly extracts the minimum from both (-4 for f from vertex calculation, -2 for g from scanning table) and compares accurately, showing f has the smaller minimum value. A common distractor like choice D might misinterpret 'larger minimum' by confusing inequality, but since -4 < -2, f's is smaller; focus on the numerical comparison. Property extraction by representation type: FROM FORMULA—y-intercept: set x = 0; maximum of quadratic: complete square or use vertex formula; slope: read from y = mx + b or calculate rise/run. FROM GRAPH—intercepts: see where crosses axes; maximum: find highest point and read coordinates; slope: pick two points, calculate rise/run. FROM TABLE—intercept: find row where x = 0 or y = 0; maximum: scan for largest y-value; slope (linear): calculate Δy/Δx between any two points. Use the method matching your representation! Quick comparison shortcuts: for y-intercepts, formulas are fastest (plug in x = 0), but graphs let you just read off where it crosses the y-axis. For maxima, graphs are easiest (visually find highest point), but formulas of quadratics give exact vertex. For growth rates, tables let you calculate differences (linear) or ratios (exponential) directly. Play to each representation's strengths—don't convert everything to formulas if the feature is obvious in the given form! Efficient comparison means using the representation smartly.

Question 4

Function ff is shown on the coordinate plane as a line passing through the points (−2,−1)(-2,-1) and (2,7)(2,7). Function gg is given algebraically by g(x)=3x+1g(x)=3x+1. Which function has the greater rate of change (slope)?

  1. Function ff has the greater slope (2 vs. 3).
  2. Function gg has the greater slope (3 vs. 2). (correct answer)
  3. Function ff has the greater slope (8 vs. 3).
  4. They have equal slopes (both 2).
Explanation: This question tests your ability to compare function properties when functions are presented in different ways—like one given graphically via points and another as a formula—requiring you to extract the same feature from each representation and compare them. Functions can be represented in four main ways, and each representation makes certain features easy to see: formulas let you calculate any value precisely and see structure (like y = mx + b shows slope m immediately), graphs show shape and extrema visually (you can spot maximums and end behavior at a glance), tables provide exact input-output pairs (easy to read specific values), and verbal descriptions summarize key features in words. To compare functions in different representations, extract the desired property from each using the method that fits that representation, then compare the extracted values! For function f shown as a line through (-2,-1) and (2,7), calculate slope as (7 - (-1))/(2 - (-2)) = 8/4 = 2; for function g given by g(x) = 3x + 1, the slope is the coefficient of x, which is 3, so comparing 3 > 2 shows g has the greater slope. Choice B correctly extracts the slope from both and compares accurately, showing g has the greater slope (3 vs. 2). A distractor like choice A might result from reversing the comparison after calculation, incorrectly claiming f has the greater slope (2 vs. 3), but ensure you identify which value is actually larger. Property extraction by representation type: FROM FORMULA—y-intercept: set x = 0; maximum of quadratic: complete square or use vertex formula; slope: read from y = mx + b or calculate rise/run. FROM GRAPH—intercepts: see where crosses axes; maximum: find highest point and read coordinates; slope: pick two points, calculate rise/run. FROM TABLE—intercept: find row where x = 0 or y = 0; maximum: scan for largest y-value; slope (linear): calculate Δy/Δx between any two points. Use the method matching your representation! Quick comparison shortcuts: for y-intercepts, formulas are fastest (plug in x = 0), but graphs let you just read off where it crosses the y-axis. For maxima, graphs are easiest (visually find highest point), but formulas of quadratics give exact vertex. For growth rates, tables let you calculate differences (linear) or ratios (exponential) directly. Play to each representation's strengths—don't convert everything to formulas if the feature is obvious in the given form! Efficient comparison means using the representation smartly.

Question 5

Function ff is given by f(x)=(13)xf(x)=\left(\frac{1}{3}\right)^x. Function gg is described verbally as: "An exponential function that starts at g(0)=2g(0)=2 and triples each time xx increases by 1." Which function has the larger output at x=2x=2?

  1. Function ff
  2. Function gg (correct answer)
  3. They are equal at x=2x=2.
  4. Not enough information to compare f(2)f(2) and g(2)g(2).
Explanation: This question tests your ability to compare function properties when functions are presented in different ways—like one given as a formula and another as a graph—requiring you to extract the same feature from each representation and compare them. Functions can be represented in four main ways, and each representation makes certain features easy to see: formulas let you calculate any value precisely and see structure (like y = mx + b shows slope m immediately), graphs show shape and extrema visually (you can spot maximums and end behavior at a glance), tables provide exact input-output pairs (easy to read specific values), and verbal descriptions summarize key features in words. To compare functions in different representations, extract the desired property from each using the method that fits that representation, then compare the extracted values! For function f given by f(x) = (1/3)^x, we calculate f(2) = (1/3)² = 1/9 ≈ 0.111. For function g described as starting at g(0) = 2 and tripling each time x increases by 1, we have g(1) = 2×3 = 6 and g(2) = 6×3 = 18. Choice B correctly identifies that g has the larger output at x = 2 since 18 > 1/9. A common error would be to misinterpret "triples" as multiplying by 1/3 instead of by 3, or to confuse the growth pattern of g with that of f, but g grows by a factor of 3 while f decays by a factor of 1/3. Property extraction by representation type: FROM FORMULA—y-intercept: set x = 0; maximum of quadratic: complete square or use vertex formula; slope: read from y = mx + b or calculate rise/run. FROM GRAPH—intercepts: see where crosses axes; maximum: find highest point and read coordinates; slope: pick two points, calculate rise/run. FROM TABLE—intercept: find row where x = 0 or y = 0; maximum: scan for largest y-value; slope (linear): calculate Δy/Δx between any two points. Use the method matching your representation! Quick comparison shortcuts: for y-intercepts, formulas are fastest (plug in x = 0), but graphs let you just read off where it crosses the y-axis. For maxima, graphs are easiest (visually find highest point), but formulas of quadratics give exact vertex. For growth rates, tables let you calculate differences (linear) or ratios (exponential) directly. Play to each representation's strengths—don't convert everything to formulas if the feature is obvious in the given form! Efficient comparison means using the representation smartly.

Question 6

Function ff is given by f(x)=x+2x−1f(x)=\dfrac{x+2}{x-1}. Function gg is described verbally as: "A rational function with a vertical asymptote at x=−4x=-4." Which function has the vertical asymptote farther from the origin (i.e., larger ∣x∣|x| value)?

  1. Function ff (its vertical asymptote is farther).
  2. Function gg (its vertical asymptote is farther). (correct answer)
  3. They are equally far from the origin.
  4. Neither function has a vertical asymptote.
Explanation: This question tests your ability to compare function properties when functions are presented in different ways—like one given as a formula and another as a graph—requiring you to extract the same feature from each representation and compare them. Functions can be represented in four main ways, and each representation makes certain features easy to see: formulas let you calculate any value precisely and see structure (like y = mx + b shows slope m immediately), graphs show shape and extrema visually (you can spot maximums and end behavior at a glance), tables provide exact input-output pairs (easy to read specific values), and verbal descriptions summarize key features in words. To compare functions in different representations, extract the desired property from each using the method that fits that representation, then compare the extracted values! For function f given by f(x) = (x+2)/(x-1), the vertical asymptote occurs where the denominator equals zero: x - 1 = 0, so x = 1, which is distance |1| = 1 from the origin. Function g is described as having a vertical asymptote at x = -4, which is distance |-4| = 4 from the origin. Choice B correctly identifies that g has the vertical asymptote farther from the origin since 4 > 1. A common mistake would be to compare -4 and 1 directly without taking absolute values, but distance from the origin is always measured as the absolute value of the coordinate. Property extraction by representation type: FROM FORMULA—y-intercept: set x = 0; maximum of quadratic: complete square or use vertex formula; slope: read from y = mx + b or calculate rise/run. FROM GRAPH—intercepts: see where crosses axes; maximum: find highest point and read coordinates; slope: pick two points, calculate rise/run. FROM TABLE—intercept: find row where x = 0 or y = 0; maximum: scan for largest y-value; slope (linear): calculate Δy/Δx between any two points. Use the method matching your representation! Quick comparison shortcuts: for y-intercepts, formulas are fastest (plug in x = 0), but graphs let you just read off where it crosses the y-axis. For maxima, graphs are easiest (visually find highest point), but formulas of quadratics give exact vertex. For growth rates, tables let you calculate differences (linear) or ratios (exponential) directly. Play to each representation's strengths—don't convert everything to formulas if the feature is obvious in the given form! Efficient comparison means using the representation smartly.

Question 7

Function ff is given by the equation f(x)=4⋅3xf(x)=4\cdot 3^x. Function gg is described verbally as: "An exponential function with initial value g(0)=6g(0)=6 and growth factor 22 (so each time xx increases by 1, the output is multiplied by 2)."

Which function has the faster growth rate?

  1. Function gg, because its initial value is larger.
  2. Function ff, because its growth factor is 33 compared to 22. (correct answer)
  3. Function gg, because its growth factor is 22 compared to 33.
  4. They have the same growth rate.
Explanation: This question tests your ability to compare function properties when functions are presented in different ways—like one given as a formula and another as a verbal description—requiring you to extract the same feature from each representation and compare them. Functions can be represented in four main ways, and each representation makes certain features easy to see: formulas let you calculate any value precisely and see structure (like y = mx + b shows slope m immediately), graphs show shape and extrema visually (you can spot maximums and end behavior at a glance), tables provide exact input-output pairs (easy to read specific values), and verbal descriptions summarize key features in words. To compare functions in different representations, extract the desired property from each using the method that fits that representation, then compare the extracted values! For function f, given as f(x) = 4 * 3^x, the growth factor is 3 (multiplier per unit x increase). For function g, verbal description states growth factor 2, so f's 3 > 2 means faster growth. Choice B correctly extracts the growth factors (3 for f from exponential base, 2 for g from description) and compares accurately, showing f has the faster growth rate. A common distractor like choice C might reverse the comparison, but since 3 > 2, f grows faster; initial values don't affect the rate itself. Property extraction by representation type: FROM FORMULA—y-intercept: set x = 0; maximum of quadratic: complete square or use vertex formula; slope: read from y = mx + b or calculate rise/run. FROM GRAPH—intercepts: see where crosses axes; maximum: find highest point and read coordinates; slope: pick two points, calculate rise/run. FROM TABLE—intercept: find row where x = 0 or y = 0; maximum: scan for largest y-value; slope (linear): calculate Δy/Δx between any two points. Use the method matching your representation! Quick comparison shortcuts: for y-intercepts, formulas are fastest (plug in x = 0), but graphs let you just read off where it crosses the y-axis. For maxima, graphs are easiest (visually find highest point), but formulas of quadratics give exact vertex. For growth rates, tables let you calculate differences (linear) or ratios (exponential) directly. Play to each representation's strengths—don't convert everything to formulas if the feature is obvious in the given form! Efficient comparison means using the representation smartly.

Question 8

Function ff is given by f(x)=(12)xf(x)=\left(\dfrac{1}{2}\right)^x. Function gg is represented by the table:

x0123g(x)5102040\begin{array}{c|cccc} x & 0 & 1 & 2 & 3\\ \hline g(x) & 5 & 10 & 20 & 40 \end{array}

Which function has the greater output at x=3x=3?

  1. Not enough information to compare at x=3x=3.
  2. They are equal at x=3x=3.
  3. Function ff, because f(3)=18f(3)=\dfrac{1}{8} is greater than 4040.
  4. Function gg, because g(3)=40g(3)=40 is greater than f(3)=18f(3)=\dfrac{1}{8}. (correct answer)
Explanation: This question tests your ability to compare function properties when functions are presented in different ways—like one given as a formula and another as a table—requiring you to extract the same feature from each representation and compare them. Functions can be represented in four main ways, and each representation makes certain features easy to see: formulas let you calculate any value precisely and see structure (like y = mx + b shows slope m immediately), graphs show shape and extrema visually (you can spot maximums and end behavior at a glance), tables provide exact input-output pairs (easy to read specific values), and verbal descriptions summarize key features in words. To compare functions in different representations, extract the desired property from each using the method that fits that representation, then compare the extracted values! For function f, given as f(x) = (1/2)^x, calculate f(3) = (1/2)^3 = 1/8. For function g, read from table g(3) = 40 directly. Choice B correctly extracts the output at x=3 from both (1/8 for f by exponentiation, 40 for g from table) and compares accurately, showing g has the greater output. A common distractor like choice A might swap the values or miscalculate (1/2)^3 as something larger, but remember it's 1/8, and 40 > 1/8 is clear. Property extraction by representation type: FROM FORMULA—y-intercept: set x = 0; maximum of quadratic: complete square or use vertex formula; slope: read from y = mx + b or calculate rise/run. FROM GRAPH—intercepts: see where crosses axes; maximum: find highest point and read coordinates; slope: pick two points, calculate rise/run. FROM TABLE—intercept: find row where x = 0 or y = 0; maximum: scan for largest y-value; slope (linear): calculate Δy/Δx between any two points. Use the method matching your representation! Quick comparison shortcuts: for y-intercepts, formulas are fastest (plug in x = 0), but graphs let you just read off where it crosses the y-axis. For maxima, graphs are easiest (visually find highest point), but formulas of quadratics give exact vertex. For growth rates, tables let you calculate differences (linear) or ratios (exponential) directly. Play to each representation's strengths—don't convert everything to formulas if the feature is obvious in the given form! Efficient comparison means using the representation smartly.

Question 9

Function ff is given by f(x)=x+4x−1f(x)=\dfrac{x+4}{x-1}. Function gg is described verbally as: "gg is a rational function with a vertical asymptote at x=−3x=-3." Which function has a vertical asymptote farther from the origin (i.e., with larger ∣x∣|x|)?

  1. Function ff (at x=1x=1) is farther from the origin than function gg (at x=−3x=-3).
  2. Function gg (at x=−3x=-3) is farther from the origin than function ff (at x=1x=1). (correct answer)
  3. They are equally far from the origin (both at distance 3).
  4. They are equally far from the origin (both at distance 1).
Explanation: This question tests your ability to compare function properties when functions are presented in different ways—like one given as a formula and another verbally—requiring you to extract the same feature from each representation and compare them. Functions can be represented in four main ways, and each representation makes certain features easy to see: formulas let you calculate any value precisely and see structure (like y = mx + b shows slope m immediately), graphs show shape and extrema visually (you can spot maximums and end behavior at a glance), tables provide exact input-output pairs (easy to read specific values), and verbal descriptions summarize key features in words. To compare functions in different representations, extract the desired property from each using the method that fits that representation, then compare the extracted values! For function f given by f(x) = (x+4)/(x-1), the vertical asymptote is where the denominator is zero, at x=1, so |x|=1; for function g described verbally as having a vertical asymptote at x=-3, |x|=3, so comparing 3 > 1 shows g's asymptote is farther from the origin. Choice B correctly extracts the asymptote positions from both and compares accurately, showing g (at x=-3) is farther from the origin than f (at x=1). A distractor like choice A might stem from forgetting to take absolute values or miscomparing distances, incorrectly stating f is farther, but always compute |x| for distance from origin. Property extraction by representation type: FROM FORMULA—y-intercept: set x = 0; maximum of quadratic: complete square or use vertex formula; slope: read from y = mx + b or calculate rise/run. FROM GRAPH—intercepts: see where crosses axes; maximum: find highest point and read coordinates; slope: pick two points, calculate rise/run. FROM TABLE—intercept: find row where x = 0 or y = 0; maximum: scan for largest y-value; slope (linear): calculate Δy/Δx between any two points. Use the method matching your representation! Quick comparison shortcuts: for y-intercepts, formulas are fastest (plug in x = 0), but graphs let you just read off where it crosses the y-axis. For maxima, graphs are easiest (visually find highest point), but formulas of quadratics give exact vertex. For growth rates, tables let you calculate differences (linear) or ratios (exponential) directly. Play to each representation's strengths—don't convert everything to formulas if the feature is obvious in the given form! Efficient comparison means using the representation smartly.

Question 10

Function ff is given by f(x)=−x4+2x2f(x)=-x^4+2x^2. Function gg is represented graphically as a polynomial whose ends both rise (as x→±∞x\to\pm\infty, g(x)→∞g(x)\to\infty). Which statement correctly compares the end behavior of ff and gg?

  1. Both ff and gg rise on the left and right.
  2. ff falls to the left and rises to the right, while gg rises on both ends.
  3. ff rises on both ends, while gg falls on both ends.
  4. ff falls on both ends, while gg rises on both ends. (correct answer)
Explanation: This question tests your ability to compare function properties when functions are presented in different ways—like one given as a formula and another graphically—requiring you to extract the same feature from each representation and compare them. Functions can be represented in four main ways, and each representation makes certain features easy to see: formulas let you calculate any value precisely and see structure (like y = mx + b shows slope m immediately), graphs show shape and extrema visually (you can spot maximums and end behavior at a glance), tables provide exact input-output pairs (easy to read specific values), and verbal descriptions summarize key features in words. To compare functions in different representations, extract the desired property from each using the method that fits that representation, then compare the extracted values! For function f given by f(x) = -x^4 + 2x^2, the leading term -x^4 indicates as x→±∞, f→-∞, so falls on both ends; for function g represented graphically with both ends rising (as x→±∞, g→∞), it rises on both ends, so they differ in end behavior. Choice B correctly extracts the end behavior from both and compares accurately, showing f falls on both ends, while g rises on both ends. A distractor like choice A might ignore the negative leading coefficient for f, incorrectly stating both rise, but always check the sign and degree of the leading term. Property extraction by representation type: FROM FORMULA—y-intercept: set x = 0; maximum of quadratic: complete square or use vertex formula; slope: read from y = mx + b or calculate rise/run. FROM GRAPH—intercepts: see where crosses axes; maximum: find highest point and read coordinates; slope: pick two points, calculate rise/run. FROM TABLE—intercept: find row where x = 0 or y = 0; maximum: scan for largest y-value; slope (linear): calculate Δy/Δx between any two points. Use the method matching your representation! Quick comparison shortcuts: for y-intercepts, formulas are fastest (plug in x = 0), but graphs let you just read off where it crosses the y-axis. For maxima, graphs are easiest (visually find highest point), but formulas of quadratics give exact vertex. For growth rates, tables let you calculate differences (linear) or ratios (exponential) directly. Play to each representation's strengths—don't convert everything to formulas if the feature is obvious in the given form! Efficient comparison means using the representation smartly.

Question 11

Function ff is given in the table:

x−1012f(x)−3159\begin{array}{c|cccc} x & -1 & 0 & 1 & 2\\ \hline f(x) & -3 & 1 & 5 & 9 \end{array}

Function gg is given algebraically by g(x)=4x+1g(x)=4x+1.

Which statement correctly compares the rate of change (slope) of ff and gg?​

  1. ff has slope 44 and gg has slope 22, so ff increases faster.
  2. ff has slope 22 and gg has slope 44, so gg increases faster.
  3. Both have slope 44, so they increase at the same rate. (correct answer)
  4. Both have slope 22, so they increase at the same rate.
Explanation: This question tests your ability to compare function properties when functions are presented in different ways—like one given as a table and another as a formula—requiring you to extract the same feature from each representation and compare them. Functions can be represented in four main ways, and each representation makes certain features easy to see: formulas let you calculate any value precisely and see structure (like y = mx + b shows slope m immediately), graphs show shape and extrema visually (you can spot maximums and end behavior at a glance), tables provide exact input-output pairs (easy to read specific values), and verbal descriptions summarize key features in words. To compare functions in different representations, extract the desired property from each using the method that fits that representation, then compare the extracted values! For function f, from table, calculate slope as Δy/Δx, e.g., (1 - (-3))/(0 - (-1)) = 4/1 = 4, consistent between points. For function g, given as g(x) = 4x + 1, slope is 4, so equal. Choice C correctly extracts the slopes (4 for f by differences in table, 4 for g from coefficient) and compares accurately, showing they have the same rate of change. A common distractor like choice A might miscalculate f's slope, perhaps using wrong points or arithmetic, but verify multiple intervals: all give 4, matching g's 4. Property extraction by representation type: FROM FORMULA—y-intercept: set x = 0; maximum of quadratic: complete square or use vertex formula; slope: read from y = mx + b or calculate rise/run. FROM GRAPH—intercepts: see where crosses axes; maximum: find highest point and read coordinates; slope: pick two points, calculate rise/run. FROM TABLE—intercept: find row where x = 0 or y = 0; maximum: scan for largest y-value; slope (linear): calculate Δy/Δx between any two points. Use the method matching your representation! Quick comparison shortcuts: for y-intercepts, formulas are fastest (plug in x = 0), but graphs let you just read off where it crosses the y-axis. For maxima, graphs are easiest (visually find highest point), but formulas of quadratics give exact vertex. For growth rates, tables let you calculate differences (linear) or ratios (exponential) directly. Play to each representation's strengths—don't convert everything to formulas if the feature is obvious in the given form! Efficient comparison means using the representation smartly.

Question 12

Function ff is given algebraically by f(x)=(x−2)(x+1)f(x)=(x-2)(x+1). Function gg is shown on the coordinate plane as a curve that crosses the x-axis at x=−4x=-4 and x=3x=3.

Which function has the greater number of x-intercepts?​

  1. They have the same number of x-intercepts (2 each). (correct answer)
  2. They have the same number of x-intercepts (1 each).
  3. Function ff has more x-intercepts (2 vs 1).
  4. Function gg has more x-intercepts (2 vs 1).
Explanation: This question tests your ability to compare function properties when functions are presented in different ways—like one given as a formula and another as a graph—requiring you to extract the same feature from each representation and compare them. Functions can be represented in four main ways, and each representation makes certain features easy to see: formulas let you calculate any value precisely and see structure (like y = mx + b shows slope m immediately), graphs show shape and extrema visually (you can spot maximums and end behavior at a glance), tables provide exact input-output pairs (easy to read specific values), and verbal descriptions summarize key features in words. To compare functions in different representations, extract the desired property from each using the method that fits that representation, then compare the extracted values! For function f, given as f(x) = (x-2)(x+1), x-intercepts are roots x=2 and x=-1, so two. For function g, graph description says crosses x-axis at x=-4 and x=3, so also two intercepts. Choice C correctly extracts the number of x-intercepts from both (2 for f by factoring, 2 for g by counting crossings) and compares accurately, showing they have the same number. A common distractor like choice A might miscount f's roots, perhaps overlooking the factored form shows two distinct, but verify by setting to zero: x=2 and x=-1. Property extraction by representation type: FROM FORMULA—y-intercept: set x = 0; maximum of quadratic: complete square or use vertex formula; slope: read from y = mx + b or calculate rise/run. FROM GRAPH—intercepts: see where crosses axes; maximum: find highest point and read coordinates; slope: pick two points, calculate rise/run. FROM TABLE—intercept: find row where x = 0 or y = 0; maximum: scan for largest y-value; slope (linear): calculate Δy/Δx between any two points. Use the method matching your representation! Quick comparison shortcuts: for y-intercepts, formulas are fastest (plug in x = 0), but graphs let you just read off where it crosses the y-axis. For maxima, graphs are easiest (visually find highest point), but formulas of quadratics give exact vertex. For growth rates, tables let you calculate differences (linear) or ratios (exponential) directly. Play to each representation's strengths—don't convert everything to formulas if the feature is obvious in the given form! Efficient comparison means using the representation smartly.

Question 13

Function ff is given by the equation f(x)=4⋅3xf(x)=4\cdot 3^x. Function gg is described verbally as: "An exponential function with initial value g(0)=6g(0)=6 and growth factor 22 (so each time xx increases by 1, the output is multiplied by 2)."

Which function has the faster growth rate?​

  1. They have the same growth rate.
  2. Function gg, because its initial value is larger.
  3. Function gg, because its growth factor is 22 compared to 33.
  4. Function ff, because its growth factor is 33 compared to 22. (correct answer)
Explanation: This question tests your ability to compare function properties when functions are presented in different ways—like one given as a formula and another as a verbal description—requiring you to extract the same feature from each representation and compare them. Functions can be represented in four main ways, and each representation makes certain features easy to see: formulas let you calculate any value precisely and see structure (like y = mx + b shows slope m immediately), graphs show shape and extrema visually (you can spot maximums and end behavior at a glance), tables provide exact input-output pairs (easy to read specific values), and verbal descriptions summarize key features in words. To compare functions in different representations, extract the desired property from each using the method that fits that representation, then compare the extracted values! For function f, given as f(x) = 4 * 3^x, the growth factor is 3 (multiplier per unit x increase). For function g, verbal description states growth factor 2, so f's 3 > 2 means faster growth. Choice B correctly extracts the growth factors (3 for f from exponential base, 2 for g from description) and compares accurately, showing f has the faster growth rate. A common distractor like choice C might reverse the comparison, but since 3 > 2, f grows faster; initial values don't affect the rate itself. Property extraction by representation type: FROM FORMULA—y-intercept: set x = 0; maximum of quadratic: complete square or use vertex formula; slope: read from y = mx + b or calculate rise/run. FROM GRAPH—intercepts: see where crosses axes; maximum: find highest point and read coordinates; slope: pick two points, calculate rise/run. FROM TABLE—intercept: find row where x = 0 or y = 0; maximum: scan for largest y-value; slope (linear): calculate Δy/Δx between any two points. Use the method matching your representation! Quick comparison shortcuts: for y-intercepts, formulas are fastest (plug in x = 0), but graphs let you just read off where it crosses the y-axis. For maxima, graphs are easiest (visually find highest point), but formulas of quadratics give exact vertex. For growth rates, tables let you calculate differences (linear) or ratios (exponential) directly. Play to each representation's strengths—don't convert everything to formulas if the feature is obvious in the given form! Efficient comparison means using the representation smartly.

Question 14

Function ff is given by the equation f(x)=2xf(x)=2^x. Function gg is represented numerically by the table:

x01234g(x)36122448\begin{array}{c|ccccc} x & 0 & 1 & 2 & 3 & 4\\ \hline g(x) & 3 & 6 & 12 & 24 & 48 \end{array}

Which function has the larger y-intercept?​

  1. Function ff has the larger y-intercept (22 vs 33).
  2. Function gg has the larger y-intercept (33 vs 11). (correct answer)
  3. Function ff has the larger y-intercept (11 vs 00).
  4. The y-intercepts are equal.
Explanation: This question tests your ability to compare function properties when functions are presented in different ways—like one given as a formula and another as a table—requiring you to extract the same feature from each representation and compare them. Functions can be represented in four main ways, and each representation makes certain features easy to see: formulas let you calculate any value precisely and see structure (like y = mx + b shows slope m immediately), graphs show shape and extrema visually (you can spot maximums and end behavior at a glance), tables provide exact input-output pairs (easy to read specific values), and verbal descriptions summarize key features in words. To compare functions in different representations, extract the desired property from each using the method that fits that representation, then compare the extracted values! For function f, given as f(x) = 2^x, find y-intercept by setting x=0: 2^0 = 1. For function g, from the table, read g(0) = 3 directly from the x=0 column. Choice B correctly extracts the y-intercept from both (1 for f by plugging in x=0, 3 for g from table) and compares accurately, showing g has the larger y-intercept. A common distractor like choice A might miscalculate f(0) as 2 instead of 1, but remember exponential with base 2 at x=0 is always 1, and verify table's x=0 value is 3. Property extraction by representation type: FROM FORMULA—y-intercept: set x = 0; maximum of quadratic: complete square or use vertex formula; slope: read from y = mx + b or calculate rise/run. FROM GRAPH—intercepts: see where crosses axes; maximum: find highest point and read coordinates; slope: pick two points, calculate rise/run. FROM TABLE—intercept: find row where x = 0 or y = 0; maximum: scan for largest y-value; slope (linear): calculate Δy/Δx between any two points. Use the method matching your representation! Quick comparison shortcuts: for y-intercepts, formulas are fastest (plug in x = 0), but graphs let you just read off where it crosses the y-axis. For maxima, graphs are easiest (visually find highest point), but formulas of quadratics give exact vertex. For growth rates, tables let you calculate differences (linear) or ratios (exponential) directly. Play to each representation's strengths—don't convert everything to formulas if the feature is obvious in the given form! Efficient comparison means using the representation smartly.

Question 15

Function ff is given algebraically by f(x)=x+4x−1f(x)=\dfrac{x+4}{x-1}. Function gg is described verbally as: "gg is a rational function with a vertical asymptote at x=−3x=-3."

Which function has a vertical asymptote farther from the origin on the x-axis?​

  1. Function ff, because its vertical asymptote is at x=4x=4.
  2. Function gg, because ∣−3∣>∣1∣|-3| > |1|. (correct answer)
  3. Function ff, because ∣1∣>∣−3∣|1| > |-3|.
  4. They are equally far from the origin.
Explanation: This question tests your ability to compare function properties when functions are presented in different ways—like one given as a formula and another as a verbal description—requiring you to extract the same feature from each representation and compare them. Functions can be represented in four main ways, and each representation makes certain features easy to see: formulas let you calculate any value precisely and see structure (like y = mx + b shows slope m immediately), graphs show shape and extrema visually (you can spot maximums and end behavior at a glance), tables provide exact input-output pairs (easy to read specific values), and verbal descriptions summarize key features in words. To compare functions in different representations, extract the desired property from each using the method that fits that representation, then compare the extracted values! For function f, given as f(x) = (x+4)/(x-1), the vertical asymptote is where denominator is zero (x=1). For function g, the verbal description directly states vertical asymptote at x=-3; compare distances |1| = 1 vs |-3| = 3, so g is farther. Choice B correctly extracts the asymptotes (x=1 from f's denominator, x=-3 from g's description) and compares distances accurately, showing g has the vertical asymptote farther from the origin. A common distractor like choice C might confuse the comparison by saying |1| > |-3|, but absolute values make both positive, and 3 > 1, so correct the inequality direction. Property extraction by representation type: FROM FORMULA—y-intercept: set x = 0; maximum of quadratic: complete square or use vertex formula; slope: read from y = mx + b or calculate rise/run. FROM GRAPH—intercepts: see where crosses axes; maximum: find highest point and read coordinates; slope: pick two points, calculate rise/run. FROM TABLE—intercept: find row where x = 0 or y = 0; maximum: scan for largest y-value; slope (linear): calculate Δy/Δx between any two points. Use the method matching your representation! Quick comparison shortcuts: for y-intercepts, formulas are fastest (plug in x = 0), but graphs let you just read off where it crosses the y-axis. For maxima, graphs are easiest (visually find highest point), but formulas of quadratics give exact vertex. For growth rates, tables let you calculate differences (linear) or ratios (exponential) directly. Play to each representation's strengths—don't convert everything to formulas if the feature is obvious in the given form! Efficient comparison means using the representation smartly.

Question 16

Function ff is given by f(x)=(12)xf(x)=\left(\dfrac{1}{2}\right)^x. Function gg is represented by the table:

x0123g(x)5102040\begin{array}{c|cccc} x & 0 & 1 & 2 & 3 \\ \hline g(x) & 5 & 10 & 20 & 40 \end{array}

Which function has the greater output at x=3x=3?

  1. Function ff, because f(3)=18f(3)=\dfrac{1}{8} is greater than 4040.
  2. Function gg, because g(3)=40g(3)=40 is greater than f(3)=18f(3)=\dfrac{1}{8}. (correct answer)
  3. They are equal at x=3x=3.
  4. Not enough information to compare at x=3x=3.
Explanation: This question tests your ability to compare function properties when functions are presented in different ways—like one given as a formula and another as a table—requiring you to extract the same feature from each representation and compare them. Functions can be represented in four main ways, and each representation makes certain features easy to see: formulas let you calculate any value precisely and see structure (like y=mx+by = mx + b shows slope mm immediately), graphs show shape and extrema visually (you can spot maximums and end behavior at a glance), tables provide exact input-output pairs (easy to read specific values), and verbal descriptions summarize key features in words. To compare functions in different representations, extract the desired property from each using the method that fits that representation, then compare the extracted values! For function ff, given as f(x)=(12)xf(x) = \left(\dfrac{1}{2}\right)^x, calculate f(3)=(12)3=18f(3) = \left(\dfrac{1}{2}\right)^3 = \dfrac{1}{8}. For function gg, read from table g(3)=40g(3) = 40 directly. Choice B correctly extracts the output at x=3x=3 from both (18\dfrac{1}{8} for ff by exponentiation, 40 for gg from table) and compares accurately, showing gg has the greater output. A common distractor like choice A might swap the values or miscalculate (12)3\left(\dfrac{1}{2}\right)^3 as something larger, but remember it's 18\dfrac{1}{8}, and 40 > 18\dfrac{1}{8} is clear. Property extraction by representation type: FROM FORMULA—y-intercept: set x=0x = 0; maximum of quadratic: complete square or use vertex formula; slope: read from y=mx+by = mx + b or calculate rise/run. FROM GRAPH—intercepts: see where crosses axes; maximum: find highest point and read coordinates; slope: pick two points, calculate rise/run. FROM TABLE—intercept: find row where x=0x = 0 or y=0y = 0; maximum: scan for largest y-value; slope (linear): calculate Δy/Δx\Delta y / \Delta x between any two points. Use the method matching your representation! Quick comparison shortcuts: for y-intercepts, formulas are fastest (plug in x=0x = 0), but graphs let you just read off where it crosses the y-axis. For maxima, graphs are easiest (visually find highest point), but formulas of quadratics give exact vertex. For growth rates, tables let you calculate differences (linear) or ratios (exponential) directly. Play to each representation's strengths—don't convert everything to formulas if the feature is obvious in the given form! Efficient comparison means using the representation smartly.

Question 17

Function ff is given by f(x)=x2−6x+5f(x)=x^2-6x+5. Function gg is shown in the table:

x01234g(x)20−202\begin{array}{c|ccccc} x & 0 & 1 & 2 & 3 & 4\\ \hline g(x) & 2 & 0 & -2 & 0 & 2 \end{array}

Which statement correctly compares the minimum values of ff and gg?​

  1. ff has a smaller minimum value than gg (−4-4 vs −2-2). (correct answer)
  2. They have the same minimum value (−2-2).
  3. ff has a larger minimum value than gg (−4-4 vs −2-2).
  4. gg has a smaller minimum value than ff (−2-2 vs −1-1).
Explanation: This question tests your ability to compare function properties when functions are presented in different ways—like one given as a formula and another as a table—requiring you to extract the same feature from each representation and compare them. Functions can be represented in four main ways, and each representation makes certain features easy to see: formulas let you calculate any value precisely and see structure (like y = mx + b shows slope m immediately), graphs show shape and extrema visually (you can spot maximums and end behavior at a glance), tables provide exact input-output pairs (easy to read specific values), and verbal descriptions summarize key features in words. To compare functions in different representations, extract the desired property from each using the method that fits that representation, then compare the extracted values! For function f, given as f(x) = x^2 - 6x + 5, find minimum by vertex formula x = -b/(2a) = 6/2 = 3, then f(3) = 9 - 18 + 5 = -4. For function g, scan table values (2, 0, -2, 0, 2) to find smallest is -2. Choice A correctly extracts the minimum from both (-4 for f from vertex calculation, -2 for g from scanning table) and compares accurately, showing f has the smaller minimum value. A common distractor like choice D might misinterpret 'larger minimum' by confusing inequality, but since -4 < -2, f's is smaller; focus on the numerical comparison. Property extraction by representation type: FROM FORMULA—y-intercept: set x = 0; maximum of quadratic: complete square or use vertex formula; slope: read from y = mx + b or calculate rise/run. FROM GRAPH—intercepts: see where crosses axes; maximum: find highest point and read coordinates; slope: pick two points, calculate rise/run. FROM TABLE—intercept: find row where x = 0 or y = 0; maximum: scan for largest y-value; slope (linear): calculate Δy/Δx between any two points. Use the method matching your representation! Quick comparison shortcuts: for y-intercepts, formulas are fastest (plug in x = 0), but graphs let you just read off where it crosses the y-axis. For maxima, graphs are easiest (visually find highest point), but formulas of quadratics give exact vertex. For growth rates, tables let you calculate differences (linear) or ratios (exponential) directly. Play to each representation's strengths—don't convert everything to formulas if the feature is obvious in the given form! Efficient comparison means using the representation smartly.

Question 18

Function ff is given in the table:

x−1012f(x)−3159\begin{array}{c|cccc} x & -1 & 0 & 1 & 2\\ \hline f(x) & -3 & 1 & 5 & 9 \end{array}

Function gg is given algebraically by g(x)=4x+1g(x)=4x+1.

Which statement correctly compares the rate of change (slope) of ff and gg?

  1. ff has slope 44 and gg has slope 22, so ff increases faster.
  2. ff has slope 22 and gg has slope 44, so gg increases faster.
  3. Both have slope 44, so they increase at the same rate. (correct answer)
  4. Both have slope 22, so they increase at the same rate.
Explanation: This question tests your ability to compare function properties when functions are presented in different ways—like one given as a table and another as a formula—requiring you to extract the same feature from each representation and compare them. Functions can be represented in four main ways, and each representation makes certain features easy to see: formulas let you calculate any value precisely and see structure (like y = mx + b shows slope m immediately), graphs show shape and extrema visually (you can spot maximums and end behavior at a glance), tables provide exact input-output pairs (easy to read specific values), and verbal descriptions summarize key features in words. To compare functions in different representations, extract the desired property from each using the method that fits that representation, then compare the extracted values! For function f, from table, calculate slope as Δy/Δx, e.g., (1 - (-3))/(0 - (-1)) = 4/1 = 4, consistent between points. For function g, given as g(x) = 4x + 1, slope is 4, so equal. Choice C correctly extracts the slopes (4 for f by differences in table, 4 for g from coefficient) and compares accurately, showing they have the same rate of change. A common distractor like choice A might miscalculate f's slope, perhaps using wrong points or arithmetic, but verify multiple intervals: all give 4, matching g's 4. Property extraction by representation type: FROM FORMULA—y-intercept: set x = 0; maximum of quadratic: complete square or use vertex formula; slope: read from y = mx + b or calculate rise/run. FROM GRAPH—intercepts: see where crosses axes; maximum: find highest point and read coordinates; slope: pick two points, calculate rise/run. FROM TABLE—intercept: find row where x = 0 or y = 0; maximum: scan for largest y-value; slope (linear): calculate Δy/Δx between any two points. Use the method matching your representation! Quick comparison shortcuts: for y-intercepts, formulas are fastest (plug in x = 0), but graphs let you just read off where it crosses the y-axis. For maxima, graphs are easiest (visually find highest point), but formulas of quadratics give exact vertex. For growth rates, tables let you calculate differences (linear) or ratios (exponential) directly. Play to each representation's strengths—don't convert everything to formulas if the feature is obvious in the given form! Efficient comparison means using the representation smartly.

Question 19

Function ff is given by the equation f(x)=2xf(x)=2^x. Function gg is represented numerically by the table:

x01234g(x)36122448\begin{array}{c|ccccc} x & 0 & 1 & 2 & 3 & 4\\ \hline g(x) & 3 & 6 & 12 & 24 & 48 \end{array}

Which function has the larger y-intercept?

  1. Function ff has the larger y-intercept (22 vs 33).
  2. Function gg has the larger y-intercept (33 vs 11). (correct answer)
  3. Function ff has the larger y-intercept (11 vs 00).
  4. The y-intercepts are equal.
Explanation: This question tests your ability to compare function properties when functions are presented in different ways—like one given as a formula and another as a table—requiring you to extract the same feature from each representation and compare them. Functions can be represented in four main ways, and each representation makes certain features easy to see: formulas let you calculate any value precisely and see structure (like y = mx + b shows slope m immediately), graphs show shape and extrema visually (you can spot maximums and end behavior at a glance), tables provide exact input-output pairs (easy to read specific values), and verbal descriptions summarize key features in words. To compare functions in different representations, extract the desired property from each using the method that fits that representation, then compare the extracted values! For function f, given as f(x) = 2^x, find y-intercept by setting x=0: 2^0 = 1. For function g, from the table, read g(0) = 3 directly from the x=0 column. Choice B correctly extracts the y-intercept from both (1 for f by plugging in x=0, 3 for g from table) and compares accurately, showing g has the larger y-intercept. A common distractor like choice A might miscalculate f(0) as 2 instead of 1, but remember exponential with base 2 at x=0 is always 1, and verify table's x=0 value is 3. Property extraction by representation type: FROM FORMULA—y-intercept: set x = 0; maximum of quadratic: complete square or use vertex formula; slope: read from y = mx + b or calculate rise/run. FROM GRAPH—intercepts: see where crosses axes; maximum: find highest point and read coordinates; slope: pick two points, calculate rise/run. FROM TABLE—intercept: find row where x = 0 or y = 0; maximum: scan for largest y-value; slope (linear): calculate Δy/Δx between any two points. Use the method matching your representation! Quick comparison shortcuts: for y-intercepts, formulas are fastest (plug in x = 0), but graphs let you just read off where it crosses the y-axis. For maxima, graphs are easiest (visually find highest point), but formulas of quadratics give exact vertex. For growth rates, tables let you calculate differences (linear) or ratios (exponential) directly. Play to each representation's strengths—don't convert everything to formulas if the feature is obvious in the given form! Efficient comparison means using the representation smartly.

Question 20

Function ff is given by f(x)=∣x−3∣+2f(x)=|x-3|+2. Function gg is described verbally as: "gg is an absolute value function with minimum value 1 occurring at x=−2x=-2." Which function has the smaller minimum value?

  1. Function ff has the smaller minimum value (2 vs. 1).
  2. Function gg has the smaller minimum value (1 vs. 2). (correct answer)
  3. They have the same minimum value (both 2).
  4. They have the same minimum value (both 1).
Explanation: This question tests your ability to compare function properties when functions are presented in different ways—like one given as a formula and another verbally—requiring you to extract the same feature from each representation and compare them. Functions can be represented in four main ways, and each representation makes certain features easy to see: formulas let you calculate any value precisely and see structure (like y = mx + b shows slope m immediately), graphs show shape and extrema visually (you can spot maximums and end behavior at a glance), tables provide exact input-output pairs (easy to read specific values), and verbal descriptions summarize key features in words. To compare functions in different representations, extract the desired property from each using the method that fits that representation, then compare the extracted values! For function f given by f(x) = |x-3| + 2, the minimum is 2 at x=3 (where inside absolute value is zero); for function g described verbally as an absolute value function with minimum 1 at x=-2, the minimum is 1, so comparing 1 < 2 shows g has the smaller minimum. Choice B correctly extracts the minimum from both and compares accurately, showing g has the smaller minimum value (1 vs. 2). A distractor like choice A might reverse the comparison, incorrectly stating f has the smaller minimum (2 vs. 1), but verify which value is actually smaller. Property extraction by representation type: FROM FORMULA—y-intercept: set x = 0; maximum of quadratic: complete square or use vertex formula; slope: read from y = mx + b or calculate rise/run. FROM GRAPH—intercepts: see where crosses axes; maximum: find highest point and read coordinates; slope: pick two points, calculate rise/run. FROM TABLE—intercept: find row where x = 0 or y = 0; maximum: scan for largest y-value; slope (linear): calculate Δy/Δx between any two points. Use the method matching your representation! Quick comparison shortcuts: for y-intercepts, formulas are fastest (plug in x = 0), but graphs let you just read off where it crosses the y-axis. For maxima, graphs are easiest (visually find highest point), but formulas of quadratics give exact vertex. For growth rates, tables let you calculate differences (linear) or ratios (exponential) directly. Play to each representation's strengths—don't convert everything to formulas if the feature is obvious in the given form! Efficient comparison means using the representation smartly.