Algebra 2 Quiz: Constructing Linear And Exponential Functions

20 questions · exam conditions

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Constructing Linear And Exponential FunctionsQuestion 1 of 20

Determine whether the relationship is linear or exponential, then write the function. The table gives:

$x$ : 1, 2, 3, 4

$y$ : 2, 6, 18, 54

Which function models the data?

y=\frac{2}{3}\cdot 3^x

y=3x-1

y=2x+0

y=2\cdot 3^x

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Algebra 2 Quiz

Algebra 2 Quiz: Constructing Linear And Exponential Functions

Practice Constructing Linear And Exponential Functions 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 Constructing Linear And Exponential Functions, 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.

All questions

Question 1

Determine whether the relationship is linear or exponential, then write the function. The table gives:

$x$ : 1, 2, 3, 4

$y$ : 2, 6, 18, 54

Which function models the data?

$y=\frac{2}{3}\cdot 3^x$ (correct answer)
$y=3x-1$
$y=2x+0$
$y=2\cdot 3^x$

Explanation: This question tests your ability to construct linear or exponential functions from given information like points, tables, graphs, or descriptions of relationships. From a table, first determine which type: calculate differences between consecutive y-values (if constant → linear with slope = that difference), and calculate ratios (if constant → exponential with base = that ratio). This identification step is crucial—you can't construct the right function if you don't know which type it is! Once identified, extract the parameters (slope and intercept for linear, initial value and base for exponential) and write the formula. The constant ratios of 3 indicate exponential; using y=2 at x=1 gives a=2/3 (since 2 =

a·3^1

), so y=(2/3)·3^x. Choice C correctly constructs the exponential function with a=2/3 and b=3 from the table. A distractor like choice B omits the fractional a, giving y(1)=2·3=6≠2—solve for a using one point after finding b from ratios. Exponential construction strategy: (1) Find initial value: if you have x = 0 in data, that y is your a; otherwise calculate backward using the pattern, (2) Find base: divide consecutive y-values (with x differing by 1): b = y_{x+1}/y_x—should be constant for exponential, (3) Write f(x) = a·b^x, (4) Verify with all data points. Example: points (0, 100) and (1, 110) and (2, 121) → a = 100, b = 110/100 = 1.1 (check: 121/110 = 1.1 ✓), so f(x) = 100·(1.1)^x. The ratio test both identifies the type and gives you the base!

Question 2

The geometric sequence $3,\ 6,\ 12,\ 24,\dots$ defines values of a function at integer inputs with $g(1)=3$ , $g(2)=6$ , $g(3)=12$ , etc. Write an exponential function $g(x)$ that extends this pattern.

$g(x)=3\cdot 2^x$
$g(x)=6\cdot 2^{x-1}$
$g(x)=3\cdot 2^{x-1}$ (correct answer)
$g(x)=3x+3$

Explanation: This question tests your ability to construct linear or exponential functions from given information like points, tables, graphs, or descriptions of relationships. For exponential functions, find the initial value a (the y-value when x = 0, or work backward if needed) and the growth/decay factor b (divide consecutive y-values: b = y₂/y₁ when x increases by 1). Then write f(x) = a·b^x. For this geometric sequence with common ratio 2, adjust for starting at x=1: using g(1)=3= ab^{1}, but form g(x)=32^{x-1} fits as it gives 3 at x=1 and multiplies by 2 each time. Choice C correctly constructs the exponential with adjusted base to extend the sequence from x=1. Choice A uses g(x)=3*2^x, which gives 6 at x=1≠3; try shifting the exponent to match the starting point. Exponential construction strategy: (1) Find initial value: if you have x = 0 in data, that y is your a; otherwise calculate backward using the pattern, (2) Find base: divide consecutive y-values (with x differing by 1): b = y_{x+1}/y_x—should be constant for exponential, (3) Write f(x) = a·b^x, (4) Verify with all data points. Example: points (0, 100) and (1, 110) and (2, 121) → a = 100, b = 110/100 = 1.1 (check: 121/110 = 1.1 ✓), so f(x) = 100·(1.1)^x. The ratio test both identifies the type and gives you the base!

Question 3

Find the linear function $f(x)=mx+b$ with slope $m=-3$ that passes through the point $(2,5)$ . Which function is correct?

$f(x)=3x-1$
$f(x)=-3x+11$ (correct answer)
$f(x)=-3x-11$
$f(x)=-5x+2$

Explanation: This question tests your ability to construct linear or exponential functions from given information like points, tables, graphs, or descriptions of relationships. To construct a linear function from a slope and a point, use the point-slope form or substitute directly into y = mx + b to find b. We're given slope m = -3 and point (2,5). Substituting into f(x) = -3x + b: 5 = -3(2) + b, which gives us 5 = -6 + b, so b = 11. Therefore, f(x) = -3x + 11. Choice A correctly constructs f(x) = -3x + 11 with the given slope -3 and passing through point (2,5). Choice C incorrectly has b = -11 instead of b = 11, likely from a sign error when solving for b. Linear construction from slope and point recipe: (1) Start with f(x) = mx + b where m is given, (2) Substitute the given point (x₀, y₀) to get y₀ = mx₀ + b, (3) Solve for b: b = y₀ - mx₀, (4) Write the complete function. For m = -3 and (2,5): b = 5 - (-3)(2) = 5 - (-6) = 5 + 6 = 11, so f(x) = -3x + 11. Verify: f(2) = -3(2) + 11 = -6 + 11 = 5 ✓!

Question 4

On a coordinate plane, a straight line passes through the points $(-1,1)$ and $(2,10)$ . Construct the linear function $f(x)=mx+b$ that matches the line.

$f(x)=2x+3$
$f(x)=4x+2$
$f(x)=3x+4$ (correct answer)
$f(x)=3x+1$

Explanation: This question tests your ability to construct linear or exponential functions from given information like points, tables, graphs, or descriptions of relationships. To construct a linear function from two points, find the slope m = (y₂ - y₁)/(x₂ - x₁), then find the y-intercept b by substituting one point into y = mx + b and solving for b. Once you have m and b, you've got your function! For exponential functions, find the initial value a (the y-value when x = 0, or work backward if needed) and the growth/decay factor b (divide consecutive y-values: b = y₂/y₁ when x increases by 1). Then write f(x) = a·b^x. For points (-1,1) and (2,10), m=(10-1)/(2-(-1))=9/3=3, then using (-1,1): 1=3*(-1)+b gives b=4, so f(x)=3x+4. Choice B correctly constructs the linear function with m=3 and b=4 from the points. A distractor like choice A uses m=2 and b=3, but check slope: differences give 3, not 2—recalculate m carefully. Linear construction from two points recipe: (1) Find slope: m = (y₂ - y₁)/(x₂ - x₁), (2) Find y-intercept: substitute either point and m into y = mx + b, solve for b, (3) Write function: f(x) = [m]x + [b], (4) Verify: check that both original points work in your function. Example: (1, 4) and (3, 10) → m = (10-4)/(3-1) = 3, then 4 = 3(1) + b gives b = 1, so f(x) = 3x + 1. Check: f(1) = 4 ✓, f(3) = 10 ✓!

Question 5

A population of rabbits grows according to an exponential model. The population triples every 4 months. If there are currently 20 rabbits, which function gives the population after $m$ months?

$P(m) = 20 \cdot 3^m$
$P(m) = 20 \cdot 3^{m/4}$ (correct answer)
$P(m) = 20 + 3m$
$P(m) = 20 \cdot 4^{m/3}$

Explanation: Since the population triples every 4 months, we need

P(4) = 3P(0)

. The general form is

P(m) = P_0 \cdot b^{m/k}

where

k

is the time period for the growth factor

b

. Here,

P_0 = 20

b = 3

, and

k = 4

, giving

P(m) = 20 \cdot 3^{m/4}

. This ensures

P(4) = 20 \cdot 3^1 = 60

and

P(8) = 20 \cdot 3^2 = 180

. Choice A triples every month, too fast. Choice C is linear growth. Choice D uses incorrect parameters.

Question 6

A car rental company charges a flat fee plus a rate per mile. After driving 120 miles, the total cost is $68. After driving 200 miles, the total cost is $92. Which function represents the total cost $ $C(m)$ $for driving$ $m$ $ miles?

$C(m) = 0.30m + 32$ (correct answer)
$C(m) = 0.25m + 38$
$C(m) = 32m + 0.30$
$C(m) = 24m + 0.30$

Explanation: Using the two points (120, 68) and (200, 92), the slope is

\frac{92-68}{200-120} = \frac{24}{80} = 0.30

per mile. Using point-slope form with (120, 68):

C(m) = 0.30m + b

. Substituting:

68 = 0.30(120) + b

, so

68 = 36 + b

, giving

b = 32

. Therefore

C(m) = 0.30m + 32

. Choice B uses an incorrect slope calculation. Choice C reverses the coefficient and constant. Choice D uses an incorrect flat fee calculation.

Question 7

A radioactive substance has a half-life of 5 years. If there are initially 800 grams of the substance, which function models the amount remaining after $t$ years?

$A(t) = 800(0.5)^t$
$A(t) = 800(0.5)^{t/5}$ (correct answer)
$A(t) = 800(0.1)^{t/5}$
$A(t) = 800 - 160t$

Explanation: For exponential decay with half-life,

A(t) = A_0(0.5)^{t/h}

where

h

is the half-life. With

A_0 = 800

and

h = 5

, we get

A(t) = 800(0.5)^{t/5}

. This gives

A(5) = 800(0.5)^1 = 400

, which is correct for one half-life. Choice A would halve the substance every year, not every 5 years. Choice C uses the wrong base. Choice D represents linear decay, not exponential.

Question 8

Determine whether the relationship is linear or exponential, then write the function.

$x$ : 1, 2, 3, 4

$y$ : 2, 6, 18, 54

$y=2\cdot 3^{x-1}$ (correct answer)
$y=2\cdot 3^x$
$y=16x-14$
$y=3x-1$

Explanation: This question tests your ability to construct linear or exponential functions from given information like points, tables, graphs, or descriptions of relationships. From a table, first determine which type: calculate differences between consecutive y-values (if constant → linear with slope = that difference), and calculate ratios (if constant → exponential with base = that ratio). This identification step is crucial—you can't construct the right function if you don't know which type it is! Differences vary (4,12,36), but ratios are constant 3 (6/2=3, etc.), so exponential with base 3; adjusted form y=23^{x-1} fits starting at x=1 with y=2. Choice A correctly constructs the exponential with adjusted parameters from the table. Choice B uses y=23^x without shift, giving 6 at x=1≠2; incorporate the exponent shift for non-zero starting x. Exponential construction strategy: (1) Find initial value: if you have x = 0 in data, that y is your a; otherwise calculate backward using the pattern, (2) Find base: divide consecutive y-values (with x differing by 1): b = y_{x+1}/y_x—should be constant for exponential, (3) Write f(x) = a·b^x, (4) Verify with all data points. Example: points (0, 100) and (1, 110) and (2, 121) → a = 100, b = 110/100 = 1.1 (check: 121/110 = 1.1 ✓), so f(x) = 100·(1.1)^x. The ratio test both identifies the type and gives you the base!

Question 9

Write the exponential function in the form $f(x)=a\cdot b^x$ that passes through $(0,6)$ and $(2,24)$ . Which formula is correct?

$f(x)=6\cdot 2^x$ (correct answer)
$f(x)=12\cdot 2^x$
$f(x)=24\cdot 2^x$
$f(x)=6\cdot 4^x$

Explanation: This question tests your ability to construct linear or exponential functions from given information like points, tables, graphs, or descriptions of relationships. To construct an exponential function from two points, find the initial value a (the y-value when x = 0, or work backward if needed) and the growth/decay factor b (divide consecutive y-values: b = y₂/y₁ when x increases by 1). Then write f(x) = a·b^x. For the points (0, 6) and (2, 24), we already have a = 6 since that's the y-value when x = 0. To find b, we need to determine what value, when squared (since x goes from 0 to 2), transforms 6 into 24: 6·b² = 24, so b² = 4, which means b = 2. Choice A correctly constructs f(x) = 6·2^x with initial value 6 and base 2 from the given points. Choice C incorrectly uses base 4, which would give f(2) = 6·4² = 96, not 24. Exponential construction strategy: (1) Find initial value: if you have x = 0 in data, that y is your a; otherwise calculate backward using the pattern, (2) Find base: divide consecutive y-values (with x differing by 1): b = y_{x+1}/y_x—should be constant for exponential, (3) Write f(x) = a·b^x, (4) Verify with all data points. Let's verify: f(0) = 6·2⁰ = 6 ✓ and f(2) = 6·2² = 6·4 = 24 ✓!

Question 10

The geometric sequence is $5, 15, 45, 135, \dots$ where the term number is $x$ (so $x=1$ gives 5). Write an exponential function $g(x)$ that gives the $x$ th term.

$g(x)=15\cdot 3^{x-1}$
$g(x)=5\cdot 3^x$
$g(x)=3\cdot 5^{x-1}$
$g(x)=5\cdot 3^{x-1}$ (correct answer)

Explanation: This question tests your ability to construct linear or exponential functions from given information like points, tables, graphs, or descriptions of relationships. For exponential functions from geometric sequences, the common ratio is the base b, and you need to account for the fact that the sequence starts at x = 1, not x = 0. The sequence 5, 15, 45, 135, ... has a common ratio of 3 (15/5=3, 45/15=3, etc.), so b = 3. Since g(1) = 5, we have 5 = a·3¹ = 3a, which gives us a = 5/3. Therefore, g(x) = (5/3)·3^x = 5·3^(x-1). Choice C correctly constructs g(x) = 5·3^(x-1), which gives the first term 5 when x = 1 and has common ratio 3. Choice A incorrectly writes g(x) = 5·3^x, which would give g(1) = 15, not 5—this shifts the sequence by one position. Geometric sequence to exponential function strategy: (1) Find common ratio r between consecutive terms—that's your base b, (2) For sequences starting at x = 1, use g(x) = (first term)·b^(x-1), or find a such that g(1) = first term, (3) Write the function, (4) Verify all terms. Here: g(x) = 5·3^(x-1). Verify: g(1) = 5·3⁰ = 5 ✓, g(2) = 5·3¹ = 15 ✓, g(3) = 5·3² = 45 ✓, g(4) = 5·3³ = 135 ✓!

Question 11

The arithmetic sequence is $8,\ 13,\ 18,\ 23,\dots$ where the first term corresponds to $x=1$ . Write a linear function $f(x)$ that gives the $x$ th term.

$f(x)=8\cdot 5^{x-1}$
$f(x)=5x+8$
$f(x)=5x+3$ (correct answer)
$f(x)=8x+5$

Explanation: This question tests your ability to construct linear or exponential functions from given information like points, tables, graphs, or descriptions of relationships. An arithmetic sequence has a constant difference between consecutive terms, making it linear. The sequence 8, 13, 18, 23,... has a common difference of 5 (13-8=5, 18-13=5, etc.), so the slope m = 5. Since the first term (when x = 1) is 8, we need to find b: 8 = 5(1) + b, which gives us b = 3. Therefore, f(x) = 5x + 3. Choice C correctly constructs f(x) = 5x + 3 with slope 5 (the common difference) and y-intercept 3. Choice B incorrectly has y-intercept 8, which would give f(1) = 5(1) + 8 = 13, not the first term of 8. Linear construction from arithmetic sequence: (1) Find common difference d between consecutive terms—this is your slope m, (2) Use the first term and its position to find b: first_term = m(position) + b, (3) Write f(x) = mx + b, (4) Verify by checking several terms. Let's verify: f(1) = 5(1) + 3 = 8 ✓, f(2) = 5(2) + 3 = 13 ✓, f(3) = 5(3) + 3 = 18 ✓, f(4) = 5(4) + 3 = 23 ✓!

Question 12

A bacteria culture starts with 200 cells and grows by 10% each hour. Let $t$ be the number of hours. What function models the population $P(t)$ ?

$P(t)=200(0.10)^t$
$P(t)=200(1.10)^t$ (correct answer)
$P(t)=200+1.10t$
$P(t)=1.10(200)^t$

Explanation: This question tests your ability to construct linear or exponential functions from given information like points, tables, graphs, or descriptions of relationships. For exponential functions, find the initial value a (the y-value when x = 0, or work backward if needed) and the growth/decay factor b (divide consecutive y-values: b = y₂/y₁ when x increases by 1). Then write f(x) = a·b^x. The bacteria culture starts with 200 cells (when t = 0), so a = 200. Growing by 10% each hour means the population is multiplied by 1.10 each hour (100% + 10% = 110% = 1.10). This gives us base b = 1.10. Therefore, P(t) = 200(1.10)^t. Choice B correctly constructs P(t) = 200(1.10)^t with initial population 200 and growth factor 1.10 from the problem description. Choice A incorrectly uses 0.10 as the base, which would represent keeping only 10% each hour (a 90% decrease!), not growing by 10%. Exponential growth strategy: (1) Identify initial value: the starting amount when t = 0, (2) Convert percentage growth to decimal multiplier: growth of r% means multiply by (1 + r/100), (3) Write P(t) = a·b^t where a is initial value and b is growth factor, (4) Verify the pattern makes sense. For 10% growth: b = 1 + 0.10 = 1.10. After 1 hour: P(1) = 200(1.10)¹ = 220, which is indeed 200 + 10% of 200!

Question 13

An exponential function has the form $g(x)=a\cdot b^x$ and passes through $(0,5)$ and $(2,45)$ . Which function is correct?

$g(x)=45\cdot 3^x$
$g(x)=5\cdot 6^x$
$g(x)=5\cdot 3^x$ (correct answer)
$g(x)=5\cdot 9^x$

Explanation: This question tests your ability to construct linear or exponential functions from given information like points, tables, graphs, or descriptions of relationships. To construct a linear function from two points, find the slope m = (y₂ - y₁)/(x₂ - x₁), then find the y-intercept b by substituting one point into y = mx + b and solving for b. Once you have m and b, you've got your function! For exponential functions, find the initial value a (the y-value when x = 0, or work backward if needed) and the growth/decay factor b (divide consecutive y-values: b = y₂/y₁ when x increases by 1). Then write f(x) = a·b^x. For points (0,5) and (2,45), a = 5 (from x=0), then 45 = 5 * b^2 gives b^2 = 9 so b=3, yielding g(x) = 5·3^x. Choice B correctly constructs the exponential function with a=5 and b=3 from the points. A common distractor like choice A uses b=9, but that would give g(2)=5*81=405 ≠45—remember to solve for b using the second point after finding a. Exponential construction strategy: (1) Find initial value: if you have x = 0 in data, that y is your a; otherwise calculate backward using the pattern, (2) Find base: divide consecutive y-values (with x differing by 1): b = y_{x+1}/y_x—should be constant for exponential, (3) Write f(x) = a·b^x, (4) Verify with all data points. Example: points (0, 100) and (1, 110) and (2, 121) → a = 100, b = 110/100 = 1.1 (check: 121/110 = 1.1 ✓), so f(x) = 100·(1.1)^x. The ratio test both identifies the type and gives you the base!

Question 14

Construct a linear function from the arithmetic sequence $4,\ 9,\ 14,\ 19,\ \dots$ where the first term corresponds to $x=1$ (so $f(1)=4$ ). Write $f(x)$ in the form $f(x)=mx+b$ .

$f(x)=4x-1$
$f(x)=5x+4$
$f(x)=4x+5$
$f(x)=5x-1$ (correct answer)

Explanation: This question tests your ability to construct linear or exponential functions from given information like points, tables, graphs, or descriptions of relationships. To construct a linear function from two points, find the slope m = (y₂ - y₁)/(x₂ - x₁), then find the y-intercept b by substituting one point into y = mx + b and solving for b. Once you have m and b, you've got your function! For exponential functions, find the initial value a (the y-value when x = 0, or work backward if needed) and the growth/decay factor b (divide consecutive y-values: b = y₂/y₁ when x increases by 1). Then write f(x) = a·b^x. For this arithmetic sequence with common difference 5 and f(1)=4, m=5, then 4=5*1 + b gives b=-1, so f(x)=5x-1. Choice A correctly constructs the linear function with slope 5 and y-intercept -1 from the sequence. A distractor like choice D might use the wrong slope—remember the common difference is your m! Linear construction from two points recipe: (1) Find slope: m = (y₂ - y₁)/(x₂ - x₁), (2) Find y-intercept: substitute either point and m into y = mx + b, solve for b, (3) Write function: f(x) = [m]x + [b], (4) Verify: check that both original points work in your function. Example: (1, 4) and (3, 10) → m = (10-4)/(3-1) = 3, then 4 = 3(1) + b gives b = 1, so f(x) = 3x + 1. Check: f(1) = 4 ✓, f(3) = 10 ✓!

Question 15

A quantity starts at 200 and increases by 10% each time period. Let $t$ be the number of time periods. Which exponential function models the quantity?

$P(t)=200(1.10)^t$ (correct answer)
$P(t)=210(1.10)^t$
$P(t)=200+0.10t$
$P(t)=200(0.10)^t$

Explanation: This question tests your ability to construct linear or exponential functions from given information like points, tables, graphs, or descriptions of relationships. To construct an exponential function from a description, identify the initial value a and the growth/decay factor b. For a 10% increase, the growth factor is 1 + 0.10 = 1.10 (you multiply by 1.10 to get 110% of the original). The quantity starts at 200, so a = 200. After t time periods with 10% growth each period, the quantity is modeled by P(t) = 200(1.10)^t. Choice A correctly constructs P(t) = 200(1.10)^t with initial value 200 and growth factor 1.10 for 10% increase. Choice B incorrectly uses 0.10 as the base, which would cause decay to 10% of the previous value each period, not growth by 10%. Exponential growth construction: (1) Initial value a = starting quantity, (2) For r% growth, base b = 1 + r/100 (for 10% growth: b = 1.10), (3) For r% decay, base b = 1 - r/100 (for 10% decay: b = 0.90), (4) Write f(t) = a·b^t. Let's verify the pattern: P(0) = 200, P(1) = 200(1.10) = 220 (10% increase ✓), P(2) = 200(1.10)² = 242 (10% increase from 220 ✓)!

Question 16

A geometric sequence is given: $3, 6, 12, 24, \dots$ Let $g(n)$ give the $n$ th term with $g(1)=3$ . Which exponential function matches this sequence?

$g(n)=3\cdot 2^{n-1}$ (correct answer)
$g(n)=3\cdot 3^{n-1}$
$g(n)=6\cdot 2^n$
$g(n)=2n+1$

Explanation: This question tests your ability to construct linear or exponential functions from given information like points, tables, graphs, or descriptions of relationships. From a table, first determine which type: calculate differences between consecutive y-values (if constant → linear with slope = that difference), and calculate ratios (if constant → exponential with base = that ratio). This identification step is crucial—you can't construct the right function if you don't know which type it is! Once identified, extract the parameters (slope and intercept for linear, initial value and base for exponential) and write the formula. For the geometric sequence with constant ratio 2 and g(1)=3, adjust to g(n)=3·2^{n-1} to fit the exponent starting at n=1. Choice A correctly constructs the exponential function with a=3 and b=2 adjusted for n-1 from the sequence. A distractor like choice D uses b=3, but 3·3^{1-1}=3, then 3·3^1=9≠6—check the common ratio matches b. Exponential construction strategy: (1) Find initial value: if you have x = 0 in data, that y is your a; otherwise calculate backward using the pattern, (2) Find base: divide consecutive y-values (with x differing by 1): b = y_{x+1}/y_x—should be constant for exponential, (3) Write f(x) = a·b^x, (4) Verify with all data points. Example: points (0, 100) and (1, 110) and (2, 121) → a = 100, b = 110/100 = 1.1 (check: 121/110 = 1.1 ✓), so f(x) = 100·(1.1)^x. The ratio test both identifies the type and gives you the base!

Question 17

An exponential function has the form $f(x)=a\cdot b^x$ and passes through $(1,16)$ and $(3,64)$ . Which function is correct?

$f(x)=8\cdot 2^x$ (correct answer)
$f(x)=16\cdot 4^x$
$f(x)=16\cdot 2^x$
$f(x)=4\cdot 4^x$

Explanation: This question tests your ability to construct linear or exponential functions from given information like points, tables, graphs, or descriptions of relationships. For exponential functions, find the initial value a (the y-value when x = 0, or work backward if needed) and the growth/decay factor b (divide consecutive y-values: b = y₂/y₁ when x increases by 1). Then write f(x) = a·b^x. From points (1,16) and (3,64), we need to find a and b. Since x increases by 2, we have 64 = 16·b², so b² = 4, giving us b = 2. Now using point (1,16): 16 = a·2¹ = 2a, so a = 8. Therefore, f(x) = 8·2^x. Choice B correctly constructs f(x) = 8·2^x with initial value 8 and base 2 from the given points. Choice D incorrectly uses base 4 and initial value 16, perhaps from confusing b² with b. Exponential construction strategy when x₀ ≠ 0: (1) Use two points to find b: if x increases by n, then y₂/y₁ = b^n, (2) Solve for b, (3) Use either point to find a: y = a·b^x, so a = y/b^x, (4) Write f(x) = a·b^x. Here: 64/16 = b² gives b = 2, then 16 = a·2¹ gives a = 8, so f(x) = 8·2^x. Verify: f(1) = 8·2¹ = 16 ✓ and f(3) = 8·2³ = 64 ✓!

Question 18

Write the exponential function $k(x)=a\cdot b^x$ that passes through $(1,9)$ and $(3,81)$ . Which formula is correct?

$k(x)=9\cdot 3^{x}$
$k(x)=3\cdot 9^{x}$
$k(x)=3^{x+1}$ (correct answer)
$k(x)=3^{2x}$

a·b^3

)/(

a·b^1

) = b^2, so 9 = b^2, giving b = 3. Substituting back: 9 = a·3^1 = 3a, so a = 3. Therefore k(x) = 3·3^x = 3^(x+1). Choice C correctly constructs k(x) = 3^(x+1) which equals 3·3^x from the given points. Choice A gives 9·3^x which doesn't pass through (1,9), choice B reverses the values incorrectly, and choice D gives 3^(2x) which also doesn't fit the points. Exponential construction strategy: (1) Find initial value: if you have x = 0 in data, that y is your a; otherwise calculate backward using the pattern, (2) Find base: divide consecutive y-values (with x differing by 1): b = y_{x+1}/y_x—should be constant for exponential, (3) Write f(x) = a·b^x, (4) Verify with all data points. Example: points (0, 100) and (1, 110) and (2, 121) → a = 100, b = 110/100 = 1.1 (check: 121/110 = 1.1 ✓), so f(x) = 100·(1.1)^x. The ratio test both identifies the type and gives you the base!

Question 19

Construct a linear function $p(x)$ with slope $m=-3$ that passes through the point $(2,11)$ . Which formula is correct?

$p(x)=-3x+17$ (correct answer)
$p(x)=3x+5$
$p(x)=-3x+5$
$p(x)=-2x+15$

Explanation: This question tests your ability to construct linear or exponential functions from given information like points, tables, graphs, or descriptions of relationships. To construct a linear function from two points, find the slope m = (y₂ - y₁)/(x₂ - x₁), then find the y-intercept b by substituting one point into y = mx + b and solving for b. Once you have m and b, you've got your function! For exponential functions, find the initial value a (the y-value when x = 0, or work backward if needed) and the growth/decay factor b (divide consecutive y-values: b = y₂/y₁ when x increases by 1). Then write f(x) = a·b^x. Given slope m = -3 and point (2,11), we use the point-slope form or substitute into y = mx + b. Using 11 = -3(2) + b, we get 11 = -6 + b, so b = 17. Therefore, p(x) = -3x + 17. Choice A correctly constructs p(x) = -3x + 17 with slope -3 and y-intercept 17 from the given information. Choice B has the wrong sign on the slope (positive instead of negative), choice C has the correct slope but wrong y-intercept (5 instead of 17), and choice D has the wrong slope (-2 instead of -3). Linear construction from two points recipe: (1) Find slope: m = (y₂ - y₁)/(x₂ - x₁), (2) Find y-intercept: substitute either point and m into y = mx + b, solve for b, (3) Write function: f(x) = [m]x + [b], (4) Verify: check that both original points work in your function. Example: (1, 4) and (3, 10) → m = (10-4)/(3-1) = 3, then 4 = 3(1) + b gives b = 1, so f(x) = 3x + 1. Check: f(1) = 4 ✓, f(3) = 10 ✓!

Question 20

Write the exponential function $g(x)=a\cdot b^x$ that passes through $(0,5)$ and $(3,40)$ . Which formula is correct?

$g(x)=5\cdot 2^x$ (correct answer)
$g(x)=5\cdot 8^x$
$g(x)=8\cdot 5^x$
$g(x)=5\cdot (1.5)^x$

Explanation: This question tests your ability to construct linear or exponential functions from given information like points, tables, graphs, or descriptions of relationships. To construct a linear function from two points, find the slope m = (y₂ - y₁)/(x₂ - x₁), then find the y-intercept b by substituting one point into y = mx + b and solving for b. Once you have m and b, you've got your function! For exponential functions, find the initial value a (the y-value when x = 0, or work backward if needed) and the growth/decay factor b (divide consecutive y-values: b = y₂/y₁ when x increases by 1). Then write f(x) = a·b^x. For an exponential function g(x) = a·b^x passing through (0,5) and (3,40), we first identify a = 5 since when x = 0, g(0) = a·b⁰ = a·1 = a = 5. Now substitute the second point: 40 = 5·b³, so b³ = 8, which means b = 2. Therefore, g(x) = 5·2^x. Choice A correctly constructs g(x) = 5·2^x with initial value 5 and base 2 from the given points. Choice B incorrectly uses base 8 instead of 2, choice C swaps the initial value and base, and choice D uses base 1.5 which doesn't satisfy the second point. Exponential construction strategy: (1) Find initial value: if you have x = 0 in data, that y is your a; otherwise calculate backward using the pattern, (2) Find base: divide consecutive y-values (with x differing by 1): b = y_{x+1}/y_x—should be constant for exponential, (3) Write f(x) = a·b^x, (4) Verify with all data points. Example: points (0, 100) and (1, 110) and (2, 121) → a = 100, b = 110/100 = 1.1 (check: 121/110 = 1.1 ✓), so f(x) = 100·(1.1)^x. The ratio test both identifies the type and gives you the base!