Comparing Functions Represented in Different Ways - Algebra 1

Function $f$ is described verbally as: "A parabola that opens upward with vertex at $(2,-5)$ ." Function $g$ is given by $g(x)=(x-2)^2-3$ . Which function has the lower minimum value?
Answer: Function $f$ , because its minimum value is $-5$ , which is lower than $-3$ . This question tests your ability to work with functions represented in different ways—like formulas, graphs, and tables—and compare their properties. Functions can be shown in multiple ways: an equation gives you a formula to calculate with, a graph shows the visual pattern, a table lists specific input-output pairs, and a verbal description explains the function in words. Each representation makes different features easy to see—graphs show maximums clearly, formulas make y-intercepts obvious (just set x = 0), and tables give you exact values to read directly. For function f described as upward parabola with vertex (2,-5), minimum is -5; for function g given by g(x)= $(x-2)^2$ -3, vertex at (2,-3) so minimum -3, compare -5 and -3. Choice B correctly identifies that function f has the lower minimum value because -5 < -3. If you chose choice A, remember lower means more negative, so -5 is below -3. When comparing functions in different forms: (1) Identify what property you're comparing, (2) Extract that property from each representation using the appropriate method (substitute for formulas, read coordinates from graphs, find values in tables), (3) Compare the extracted values. Example: to compare y-intercepts, find where x = 0 in the formula, look where the graph crosses the y-axis, or find y when x = 0 in the table! Know what each representation shows best: formulas are great for calculating specific values and seeing patterns in the equation; graphs excel at showing maximums, minimums, and overall shape; tables are perfect for finding exact values at specific points; descriptions summarize key features. Use each representation's strengths!
A bike rental shop offers two pricing functions. Plan $F$ : $F(h)=12+4h$ dollars, where $h$ is the number of hours. Plan $G$ is shown in the table: $h$ : $0,\ 1,\ 2,\ 3$ $G(h)$ : $8,\ 14,\ 20,\ 26$ Which plan is cheaper for 2 hours?
Answer: They cost the same for 2 hours. This question tests your ability to work with functions represented in different ways—like formulas, graphs, and tables—and compare their properties. Functions can be shown in multiple ways: an equation gives you a formula to calculate with, a graph shows the visual pattern, a table lists specific input-output pairs, and a verbal description explains the function in words. Each representation makes different features easy to see—graphs show maximums clearly, formulas make y-intercepts obvious (just set x = 0), and tables give you exact values to read directly. To compare costs at h=2, calculate F(2)=12+4*2=20 from the formula; read G(2)=20 from the table, so they are equal. Choice D correctly identifies that they cost the same for 2 hours. You might have misread the table and thought G(2)=26 (which is actually G(3)), but always match the exact input value—you're improving every time! When comparing functions in different forms: (1) Identify what property you're comparing, (2) Extract that property from each representation using the appropriate method (substitute for formulas, read coordinates from graphs, find values in tables), (3) Compare the extracted values. Similarly, for comparing slopes of linear functions: read m from y=mx+b, calculate rise/run from a graph, or find Δy/Δx from consecutive table entries.
Function $w(x) = 2|x - 1| + 3$ . Function $z$ is shown in the graph. Which statement correctly compares the domains and ranges of these functions?
Answer: Function $w$ has an unrestricted domain, while $z$ has a restricted domain and range Function w(x) = 2|x-1| + 3 has domain (-‚àû,‚àû) and range [3,‚àû) since absolute value functions are defined for all real numbers but have a minimum value. From the graph, z appears to be defined only for x ‚àà [-2,4] and has range [0,3], making both its domain and range restricted compared to w's unrestricted domain.
Function $f$ is defined by the equation $f(x) = -2x^2 + 8x - 3$ . Function $g$ is represented by the table shown. Which statement correctly compares the y-intercepts of these functions?
Answer: The y-intercept of $g$ is 2 units greater than the y-intercept of $f$ The y-intercept of f is f(0) = -2(0)¬≤ + 8(0) - 3 = -3. From the table, when x = 0, g(x) = -1, so the y-intercept of g is -1. Since -1 - (-3) = 2, the y-intercept of g is 2 units greater than the y-intercept of f.
Function $u$ can be described as: "A linear function that passes through points $(0, 8)$ and $(5, 3)$ ." The table represents function $v$ . At what x-value do these functions have the same output?
Answer: $x = 2$ , where both functions equal 6 Function u has slope (3-8)/(5-0) = -1 and y-intercept 8, so u(x) = -x + 8. Checking the intersection: u(2) = -2 + 8 = 6, and from the table v(2) = 6. Therefore, both functions equal 6 when x = 2.
Function $h$ is described verbally as: "a linear function with slope $-4$ and y-intercept $2$ ." Function $k$ is given by the table: $x$ : $-1,\ 0,\ 1$ $k(x)$ : $5,\ 3,\ 1$ Which function has the steeper slope (greater slope magnitude)?
Answer: Function $h$ , because $|-4|>|-2|$ . This question tests your ability to work with functions represented in different ways—like formulas, graphs, and tables—and compare their properties. Functions can be shown in multiple ways: an equation gives you a formula to calculate with, a graph shows the visual pattern, a table lists specific input-output pairs, and a verbal description explains the function in words. Each representation makes different features easy to see—graphs show maximums clearly, formulas make y-intercepts obvious (just set x = 0), and tables give you exact values to read directly. To compare slope magnitudes, note the verbal description gives h a slope of -4 (magnitude 4); for k's table, calculate slope as (3-5)/(0-(-1)) = -2 or (1-3)/(1-0) = -2 (magnitude 2), so 4 > 2. Choice A correctly identifies that function h has the steeper slope because |-4| > |-2|. If you thought k's slope was -1, that's okay—just practice calculating change in y over change in x from table points to get more comfortable. When comparing functions in different forms: (1) Identify what property you're comparing, (2) Extract that property from each representation using the appropriate method (substitute for formulas, read coordinates from graphs, find values in tables), (3) Compare the extracted values. Quick y-intercept trick: in a formula, set x=0 and calculate; on a graph, see where it crosses the y-axis; in a table, find the y-value when x=0—three different methods, same property! Similarly, for comparing slopes of linear functions: read m from y=mx+b, calculate rise/run from a graph, or find Δy/Δx from consecutive table entries.
Function $f$ is given algebraically by $f(x)=2x-3$ . Function $g$ is shown by the table below. Table for $g$ : - when $x=0$ , $g(x)=1$ - when $x=2$ , $g(x)=5$ Which function has the larger y-intercept?
Answer: Function $g$ , because its y-intercept is $1$ and $1>-3$ . This question tests your ability to work with functions represented in different ways—like formulas, graphs, and tables—and compare their properties. Functions can be shown in multiple ways: an equation gives you a formula to calculate with, a graph shows the visual pattern, a table lists specific input-output pairs, and a verbal description explains the function in words. Each representation makes different features easy to see—graphs show maximums clearly, formulas make y-intercepts obvious (just set x = 0), and tables give you exact values to read directly. To compare y-intercepts, calculate f(0) = 2*0 - 3 = -3 from the formula for f, and read g(0) = 1 directly from the table for g, showing that 1 > -3. Choice B correctly identifies that function g has the larger y-intercept because 1 > -3. Don't worry if you mixed up the intercept with another point like (2,5)—just remember the y-intercept is always at x=0, so double-check that value in each representation. When comparing functions in different forms: (1) Identify what property you're comparing, (2) Extract that property from each representation using the appropriate method (substitute for formulas, read coordinates from graphs, find values in tables), (3) Compare the extracted values—for example, to compare y-intercepts, find where x=0 in the formula, look where the graph crosses the y-axis, or find y when x=0 in the table! Know what each representation shows best: formulas are great for calculating specific values and seeing patterns in the equation; graphs excel at showing maximums, minimums, and overall shape; tables are perfect for finding exact values at specific points; descriptions summarize key features—use each representation's strengths!
Function $h$ is described verbally as: "A linear function with y-intercept 6 and slope −2." Function $k$ is given by $k(x)=x+1$ . Which function has the greater rate of change (slope)?
Answer: Function $k$ has the greater slope. This question tests your ability to work with functions represented in different ways—like formulas, graphs, and tables—and compare their properties. Functions can be shown in multiple ways: an equation gives you a formula to calculate with, a graph shows the visual pattern, a table lists specific input-output pairs, and a verbal description explains the function in words. Each representation makes different features easy to see—graphs show maximums clearly, formulas make y-intercepts obvious (just set x = 0), and tables give you exact values to read directly. Function h has slope -2 (given directly in the verbal description), while function k(x) = x + 1 is in slope-intercept form y = mx + b where m = 1, so k has slope 1. Since 1 > -2, function k has the greater slope. Choice B correctly identifies that function k has the greater slope because positive 1 is greater than negative 2. If you chose A, remember that -2 is less than 1 on the number line—negative slopes go down while positive slopes go up! When comparing functions in different forms: (1) Identify what property you're comparing, (2) Extract that property from each representation using the appropriate method (read from verbal description, identify coefficient of x in formulas), (3) Compare the extracted values. Know what each representation shows best: verbal descriptions often state key features directly, while formulas in y = mx + b form show slope as the coefficient of x!
Function $c$ is described as: "A quadratic function with vertex at $(2, -1)$ that opens upward and passes through the point $(0, 3)$ ." The graph shows function $d(x)$ . Which statement about their y-intercepts and vertices is correct?
Answer: Function $c$ has a y-intercept 1 unit higher and a vertex 1 unit lower than $d$ Function c passes through (0,3) so its y-intercept is 3, and its vertex is at (2,-1). From the graph, d has y-intercept 2 and vertex at (1,0). Comparing: c's y-intercept (3) is 1 unit higher than d's (2), and c's vertex y-coordinate (-1) is 1 unit lower than d's (0).
Function $f$ is given by $f(x)=(x+1)^2-4$ . Function $g$ is described verbally as: "A parabola that opens upward and has x-intercepts at $x=-1$ and $x=3$ ." Which function has more x-intercepts?
Answer: They have the same number of x-intercepts. This question tests your ability to work with functions represented in different ways—like formulas, graphs, and tables—and compare their properties. Functions can be shown in multiple ways: an equation gives you a formula to calculate with, a graph shows the visual pattern, a table lists specific input-output pairs, and a verbal description explains the function in words. Each representation makes different features easy to see—graphs show maximums clearly, formulas make y-intercepts obvious (just set x = 0), and tables give you exact values to read directly. For f(x) = (x+1)² - 4, expand to get f(x) = x² + 2x + 1 - 4 = x² + 2x - 3. Setting f(x) = 0: x² + 2x - 3 = 0, which factors as (x+3)(x-1) = 0, giving x-intercepts at x = -3 and x = 1 (two intercepts). Function g is described as having x-intercepts at x = -1 and x = 3 (also two intercepts). Both functions have exactly 2 x-intercepts! Choice C correctly identifies that they have the same number of x-intercepts (both have 2). If you chose differently, remember to count all x-intercepts—parabolas can have 0, 1, or 2 x-intercepts depending on how they're positioned! Know what each representation shows best: verbal descriptions often list x-intercepts directly, while for formulas you need to solve f(x) = 0. Quick tip: if a parabola is given in factored form like a(x-r)(x-s), the x-intercepts are at x = r and x = s!
Function $f$ is given by $f(x)=x^2-4x+1$ . Function $g$ is given in the table. Which function has the smaller minimum value? Table for $g(x)$ : - $g(0)=5$ - $g(1)=2$ - $g(2)=1$ - $g(3)=2$ - $g(4)=5$
Answer: Function $f$ , because its minimum value is $-3$ which is less than $1$ . This question tests your ability to work with functions represented in different ways—like formulas, graphs, and tables—and compare their properties. Functions can be shown in multiple ways: an equation gives you a formula to calculate with, a graph shows the visual pattern, a table lists specific input-output pairs, and a verbal description explains the function in words. Each representation makes different features easy to see—graphs show maximums clearly, formulas make y-intercepts obvious (just set x = 0), and tables give you exact values to read directly. For function f given by f(x)= $x^2$ -4x+1, complete the square or use vertex formula x=-b/(2a)=2, then f(2)=-3 as minimum; for function g in the table, the lowest value is 1 at x=2, so compare -3 and 1. Choice A correctly identifies that function f has the smaller minimum value because -3 < 1. If you went with choice C, that's okay—just scan the table carefully for the smallest y-value. When comparing functions in different forms: (1) Identify what property you're comparing, (2) Extract that property from each representation using the appropriate method (substitute for formulas, read coordinates from graphs, find values in tables), (3) Compare the extracted values. Example: to compare y-intercepts, find where x = 0 in the formula, look where the graph crosses the y-axis, or find y when x = 0 in the table! Know what each representation shows best: formulas are great for calculating specific values and seeing patterns in the equation; graphs excel at showing maximums, minimums, and overall shape; tables are perfect for finding exact values at specific points; descriptions summarize key features. Use each representation's strengths!
Function $m$ is given by $m(x)=-2x+5$ . Function $n$ is shown by the table: $x$ : $0,\ 1,\ 2$ $n(x)$ : $5,\ 4,\ 3$ What is the difference between the y-intercepts of $m$ and $n$ (that is, $m(0)-n(0)$ )?
Answer: $0$ This question tests your ability to work with functions represented in different ways—like formulas, graphs, and tables—and compare their properties. Functions can be shown in multiple ways: an equation gives you a formula to calculate with, a graph shows the visual pattern, a table lists specific input-output pairs, and a verbal description explains the function in words. Each representation makes different features easy to see—graphs show maximums clearly, formulas make y-intercepts obvious (just set x = 0), and tables give you exact values to read directly. To find the difference in y-intercepts, calculate m(0)=-2*0+5=5 from the formula; read n(0)=5 from the table, so 5-5=0. Choice B correctly identifies that the difference is 0. It's possible to miscalculate m's intercept as -5 from the slope, but always plug in x=0 directly— you're doing awesome! When comparing functions in different forms: (1) Identify what property you're comparing, (2) Extract that property from each representation using the appropriate method (substitute for formulas, read coordinates from graphs, find values in tables), (3) Compare the extracted values. Quick y-intercept trick: in a formula, set x=0 and calculate; on a graph, see where it crosses the y-axis; in a table, find the y-value when x=0—three different methods, same property! Similarly, for comparing slopes of linear functions: read m from y=mx+b, calculate rise/run from a graph, or find Δy/Δx from consecutive table entries.
Function $A$ is given by $A(t)=50(1.10)^t$ . Function $B$ is described verbally as: "an exponential function with initial value $50$ and growth factor $1.05$ per unit of $t$ ." Which function has the faster growth rate?
Answer: Function $A$ , because $1.10>1.05$ . This question tests your ability to work with functions represented in different ways—like formulas, graphs, and tables—and compare their properties. Functions can be shown in multiple ways: an equation gives you a formula to calculate with, a graph shows the visual pattern, a table lists specific input-output pairs, and a verbal description explains the function in words. Each representation makes different features easy to see—graphs show maximums clearly, formulas make y-intercepts obvious (just set x = 0), and tables give you exact values to read directly. To compare growth rates, note A's formula has growth factor 1.10; B's verbal description gives 1.05, so 1.10 > 1.05 means A grows faster. Choice B correctly identifies that function A has the faster growth rate because 1.10 > 1.05. It's common to confuse which factor is larger, but remember the bigger the base >1, the faster the exponential growth—keep practicing! Know what each representation shows best: formulas are great for calculating specific values and seeing patterns in the equation; graphs excel at showing maximums, minimums, and overall shape; tables are perfect for finding exact values at specific points; descriptions summarize key features—use each representation's strengths! Quick y-intercept trick: in a formula, set x=0 and calculate; on a graph, see where it crosses the y-axis; in a table, find the y-value when x=0—three different methods, same property!
Function $f$ is given algebraically by $f(x)=2x-3$ . Function $g$ is given in the table. Which function has the larger y-intercept? Table for $g(x)$ : - when $x=0$ , $g(x)=1$ - when $x=2$ , $g(x)=5$
Answer: Function $g$ , because its y-intercept is $1$ and $1>-3$ . This question tests your ability to work with functions represented in different ways—like formulas, graphs, and tables—and compare their properties. Functions can be shown in multiple ways: an equation gives you a formula to calculate with, a graph shows the visual pattern, a table lists specific input-output pairs, and a verbal description explains the function in words. Each representation makes different features easy to see—graphs show maximums clearly, formulas make y-intercepts obvious (just set x = 0), and tables give you exact values to read directly. For function f given by f(x)=2x-3, the y-intercept is f(0)=-3; for function g in the table, when x=0, g(x)=1, so we compare -3 and 1 to see which is larger. Choice B correctly identifies that function g has the larger y-intercept because 1 > -3. Don't worry if you picked choice A—it's easy to mix up the signs, but remember that positive numbers are greater than negative ones. When comparing functions in different forms: (1) Identify what property you're comparing, (2) Extract that property from each representation using the appropriate method (substitute for formulas, read coordinates from graphs, find values in tables), (3) Compare the extracted values. Example: to compare y-intercepts, find where x = 0 in the formula, look where the graph crosses the y-axis, or find y when x = 0 in the table! Know what each representation shows best: formulas are great for calculating specific values and seeing patterns in the equation; graphs excel at showing maximums, minimums, and overall shape; tables are perfect for finding exact values at specific points; descriptions summarize key features. Use each representation's strengths!
Function $f$ is given by the equation $f(x)=-(x-1)^2+6$ . Function $g$ is described as: "A parabola that opens downward with vertex at $(3,4)$ ." Which function has the larger maximum value?
Answer: Function $f$ , because its maximum value is $6$ and $6>4$ . This question tests your ability to work with functions represented in different ways—like formulas, graphs, and tables—and compare their properties. Functions can be shown in multiple ways: an equation gives you a formula to calculate with, a graph shows the visual pattern, a table lists specific input-output pairs, and a verbal description explains the function in words. Each representation makes different features easy to see—graphs show maximums clearly, formulas make y-intercepts obvious (just set x = 0), and tables give you exact values to read directly. For function f given by f(x)=- $(x-1)^2$ +6, it's a downward parabola with maximum at the vertex y=6; for function g described as a downward parabola with vertex at (3,4), the maximum is 4, so we compare 6 and 4. Choice A correctly identifies that function f has the larger maximum value because 6 > 4. If you selected choice C, no problem—double-check the vertex y-coordinate in the description to confirm the maximum. When comparing functions in different forms: (1) Identify what property you're comparing, (2) Extract that property from each representation using the appropriate method (substitute for formulas, read coordinates from graphs, find values in tables), (3) Compare the extracted values. Example: to compare y-intercepts, find where x = 0 in the formula, look where the graph crosses the y-axis, or find y when x = 0 in the table! Know what each representation shows best: formulas are great for calculating specific values and seeing patterns in the equation; graphs excel at showing maximums, minimums, and overall shape; tables are perfect for finding exact values at specific points; descriptions summarize key features. Use each representation's strengths!
Function $p$ is described as follows: "The function starts at a y-value of 5 when $x = 0$ , decreases linearly to a minimum of -3 when $x = 4$ , then increases linearly back to 5 when $x = 8$ ." Function $q(x) = |x - 3| + 1$ . Which comparison of their ranges is correct?
Answer: Function $p$ has a smaller minimum value, but $q$ has no maximum value Function p has a minimum of -3 and maximum of 5, so its range is [-3, 5]. Function q(x) = |x - 3| + 1 has a minimum value of 1 (when x = 3) and no maximum since absolute value functions extend to positive infinity. Therefore, p has a smaller minimum (-3 < 1), but q has no maximum value.
The graph shows function $r(x)$ , which has x-intercepts at $x = -2$ and $x = 4$ . Function $s(x) = (x + 1)(x - 3)$ . Which statement correctly compares the x-intercepts of these functions?
Answer: The distance between the x-intercepts of $s$ is 2 units less than for $r$ Function r has x-intercepts at -2 and 4, so the distance between them is 4-(-2) = 6 units. Function s(x) = (x+1)(x-3) has x-intercepts at x = -1 and x = 3, so the distance between them is 3-(-1) = 4 units. Since 6-4 = 2, the distance between s's x-intercepts is 2 units less than r's.
Function $A$ is given by $A(t)=20(1.05)^t$ . Function $B$ is described verbally as: "An exponential function with initial value 20 that multiplies by 1.08 each time $t$ increases by 1." Which function has the faster growth rate?
Answer: Function $B$ grows faster. This question tests your ability to work with functions represented in different ways—like formulas, graphs, and tables—and compare their properties. Functions can be shown in multiple ways: an equation gives you a formula to calculate with, a graph shows the visual pattern, a table lists specific input-output pairs, and a verbal description explains the function in words. Each representation makes different features easy to see—graphs show maximums clearly, formulas make y-intercepts obvious (just set x = 0), and tables give you exact values to read directly. Function A has the form A(t) = 20 $(1.05)^t$ , where the growth factor is 1.05. Function B multiplies by 1.08 each time t increases by 1, so its growth factor is 1.08. Since 1.08 > 1.05, function B grows faster—it multiplies by a larger number each time period! Choice B correctly identifies that function B grows faster because its growth factor (1.08) is larger than A's growth factor (1.05). If you chose A, remember that in exponential functions f(t) = $a(b)^t$ , the base b determines the growth rate—larger b means faster growth! Know what each representation shows best: formulas show the growth factor as the base of the exponential, while verbal descriptions often state the multiplier directly. For exponential growth, compare the bases (growth factors) to determine which grows faster!
Function $p$ is given algebraically by $p(x)=x^2-4x+1$ . Function $q$ is described verbally as: "a parabola that opens up with vertex at $(1,-2)$ ." Which function has the smaller minimum value?
Answer: Function $p$ , because its minimum value is $-3$ , which is smaller than $-2$ . This question tests your ability to work with functions represented in different ways—like formulas, graphs, and tables—and compare their properties. Functions can be shown in multiple ways: an equation gives you a formula to calculate with, a graph shows the visual pattern, a table lists specific input-output pairs, and a verbal description explains the function in words. Each representation makes different features easy to see—graphs show maximums clearly, formulas make y-intercepts obvious (just set x = 0), and tables give you exact values to read directly. To compare minimum values, complete the square or use vertex formula for p(x) = $x^2$ - 4x + 1 to find vertex at x=2, p(2)=-3; q's verbal description gives minimum at vertex y=-2, so -3 < -2. Choice A correctly identifies that function p has the smaller minimum value because -3 is smaller than -2. If you calculated p's minimum as -1, no problem—just remember the vertex x = -b/(2a) helps find it accurately. When comparing functions in different forms: (1) Identify what property you're comparing, (2) Extract that property from each representation using the appropriate method (substitute for formulas, read coordinates from graphs, find values in tables), (3) Compare the extracted values—for example, to compare y-intercepts, find where x=0 in the formula, look where the graph crosses the y-axis, or find y when x=0 in the table! Quick y-intercept trick: in a formula, set x=0 and calculate; on a graph, see where it crosses the y-axis; in a table, find the y-value when x=0—three different methods, same property!
Function $f$ is given by $f(x)=3x+6$ . Function $g$ is shown by the table: $x$ : $-2,\ -1,\ 0,\ 1$ $g(x)$ : $4,\ 2,\ 0,\ -2$ Which function has the larger x-intercept?
Answer: Function $g$ , because its x-intercept is $0$ , which is larger than $-2$ . This question tests your ability to work with functions represented in different ways—like formulas, graphs, and tables—and compare their properties. Functions can be shown in multiple ways: an equation gives you a formula to calculate with, a graph shows the visual pattern, a table lists specific input-output pairs, and a verbal description explains the function in words. Each representation makes different features easy to see—graphs show maximums clearly, formulas make y-intercepts obvious (just set x = 0), and tables give you exact values to read directly. To compare x-intercepts, solve 3x + 6 = 0 for f to get x = -2; for g's table, note g(0) = 0, indicating x-intercept at 0 (and table suggests it's linear, crossing once), so 0 > -2. Choice B correctly identifies that function g has the larger x-intercept because 0 > -2. You might have thought f's intercept was 2 from mis-solving, but always set y=0 and solve for x carefully—you've got this! When comparing functions in different forms: (1) Identify what property you're comparing, (2) Extract that property from each representation using the appropriate method (substitute for formulas, read coordinates from graphs, find values in tables), (3) Compare the extracted values. Similarly, for comparing slopes of linear functions: read m from y=mx+b, calculate rise/run from a graph, or find Δy/Δx from consecutive table entries.

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