Algebra 2 Quiz: Constructing Linear And Exponential Functions
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Constructing Linear And Exponential FunctionsQuestion 1 of 20

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

g(x)=5⋅2xg(x)=5\cdot 2^x
g(x)=5⋅8xg(x)=5\cdot 8^x
g(x)=8⋅5xg(x)=8\cdot 5^x
g(x)=5⋅(1.5)xg(x)=5\cdot (1.5)^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

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

  1. g(x)=5⋅2xg(x)=5\cdot 2^x (correct answer)
  2. g(x)=5⋅8xg(x)=5\cdot 8^x
  3. g(x)=8⋅5xg(x)=8\cdot 5^x
  4. g(x)=5⋅(1.5)xg(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!

Question 2

An exponential function has the form g(x)=a⋅bxg(x)=a\cdot b^x and passes through (0,6)(0,6) and (2,54)(2,54). Which function is correct?

  1. g(x)=54⋅3xg(x)=54\cdot 3^x
  2. g(x)=6⋅9xg(x)=6\cdot 9^x
  3. g(x)=3⋅6xg(x)=3\cdot 6^x
  4. g(x)=6⋅3xg(x)=6\cdot 3^x (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, 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. Since the function passes through (0,6), we immediately know a = 6 because when x = 0, g(0) = a·b⁰ = a·1 = a. Now we need to find b using the point (2,54): 54 = 6·b². Solving for b: b² = 54/6 = 9, so b = 3. Therefore, g(x) = 6·3^x. Choice A correctly constructs g(x) = 6·3^x with initial value 6 and base 3 from the given points. Choice B incorrectly uses base 9, which would be b² instead of b itself. 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: g(0) = 6·3⁰ = 6 ✓ and g(2) = 6·3² = 6·9 = 54 ✓!

Question 3

Construct an exponential function from the geometric sequence 3, 12, 48, 192, …3,\ 12,\ 48,\ 192,\ \dots where the first term corresponds to x=1x=1 (so f(1)=3f(1)=3). Write f(x)f(x) in the form f(x)=a⋅bxf(x)=a\cdot b^x.

  1. f(x)=3⋅4xf(x)=3\cdot 4^x
  2. f(x)=12⋅4x−1f(x)=12\cdot 4^{x-1}
  3. f(x)=34⋅4xf(x)=\frac{3}{4}\cdot 4^x (correct answer)
  4. f(x)=3⋅3xf(x)=3\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. 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. 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 this geometric sequence with common ratio 4 and f(1)=3, solve 3 = a*4^1 gives a=3/4, so f(x)=(3/4)*4^x. Choice C correctly constructs the exponential function with coefficient 3/4 and base 4 from the sequence. A distractor like choice A might forget to adjust a for x starting at 1—use f(1) to solve for 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 4

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

xx: 1, 2, 3, 4

yy: 2, 6, 18, 54

Which function models the data?

  1. y=23⋅3xy=\frac{2}{3}\cdot 3^x (correct answer)
  2. y=3x−1y=3x-1
  3. y=2x+0y=2x+0
  4. y=2⋅3xy=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⋅31a·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 5

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

  1. f(x)=−3x+11f(x)=-3x+11 (correct answer)
  2. f(x)=3x−1f(x)=3x-1
  3. f(x)=−3x+1f(x)=-3x+1
  4. f(x)=−6x+17f(x)=-6x+17
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! Given slope m=-3 and point (2,5), substitute: 5 = -3*2 + b gives b=11, so f(x)=-3x+11. Choice A correctly constructs the linear function with m=-3 and b=11 using the given slope and point. Choice B uses positive m=3, which doesn't match the given slope—remember to use the provided m directly. 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 6

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

  1. f(x)=3x−1f(x)=3x-1
  2. f(x)=−3x+11f(x)=-3x+11 (correct answer)
  3. f(x)=−3x−11f(x)=-3x-11
  4. f(x)=−5x+2f(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 7

A water tank contains 200 liters at time t=0t=0 hours and the amount increases by 10% each hour. Write an exponential function A(t)A(t) that models the amount of water (in liters) after tt hours.

  1. A(t)=200(0.9)tA(t)=200(0.9)^t
  2. A(t)=200(1.1)tA(t)=200(1.1)^t (correct answer)
  3. A(t)=210(1.1)tA(t)=210(1.1)^t
  4. A(t)=200+10tA(t)=200+10t
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. Here, the initial amount a=200 at t=0, and it increases by 10% each hour, so growth factor b=1.1, giving A(t)=200*(1.1)^t. Choice B correctly constructs the exponential with a=200 and b=1.1 from the description. Choice A uses decay b=0.9, which would decrease the amount—remember that increase means b>1. 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 8

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

  1. g(x)=15⋅3x−1g(x)=15\cdot 3^{x-1}
  2. g(x)=5⋅3xg(x)=5\cdot 3^x
  3. g(x)=3⋅5x−1g(x)=3\cdot 5^{x-1}
  4. g(x)=5⋅3x−1g(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 9

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

  1. k(x)=9⋅3xk(x)=9\cdot 3^{x}
  2. k(x)=3⋅9xk(x)=3\cdot 9^{x}
  3. k(x)=3x+1k(x)=3^{x+1} (correct answer)
  4. k(x)=32xk(x)=3^{2x}
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 exponential k(x) = a·b^x through (1,9) and (3,81), let's set up equations: 9 = a·b^1 and 81 = a·b^3. Dividing the second by the first: 81/9 = (a⋅b3a·b^3)/(a⋅b1a·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 10

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

  1. f(x)=8⋅5x−1f(x)=8\cdot 5^{x-1}
  2. f(x)=5x+8f(x)=5x+8
  3. f(x)=5x+3f(x)=5x+3 (correct answer)
  4. f(x)=8x+5f(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 11

A savings account starts with 200200 dollars and increases by 10%10\% each month. Let tt be the number of months since the start. Write an exponential function S(t)S(t) that models the balance.

  1. S(t)=200+0.1tS(t)=200+0.1t
  2. S(t)=200(0.9)tS(t)=200(0.9)^t
  3. S(t)=200(1.1)tS(t)=200(1.1)^t (correct answer)
  4. S(t)=210(1.1)tS(t)=210(1.1)^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. 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! With initial $200 and 10% monthly growth, a=200 and b=1.1, so S(t)=200*(1.1)^t. Choice A correctly constructs the exponential function with initial value 200 and growth factor 1.1 from the description. A linear distractor like choice D might add percentages additively—remember exponential models multiplicative growth! 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 12

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

  1. g(x)=3⋅2xg(x)=3\cdot 2^x
  2. g(x)=6⋅2x−1g(x)=6\cdot 2^{x-1}
  3. g(x)=3⋅2x−1g(x)=3\cdot 2^{x-1} (correct answer)
  4. g(x)=3x+3g(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 13

Determine whether the data are best modeled by a linear or exponential function, and write a function that models the data.

Table: x:0, 1, 2, 3x: 0,\ 1,\ 2,\ 3 and f(x):5, 10, 20, 40f(x): 5,\ 10,\ 20,\ 40

  1. f(x)=5⋅2xf(x)=5\cdot 2^x (correct answer)
  2. f(x)=10⋅2xf(x)=10\cdot 2^x
  3. f(x)=5x+5f(x)=5x+5
  4. f(x)=5⋅4xf(x)=5\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. 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. Here, ratios are constantly 2, so exponential with b=2; f(0)=5 is a, so f(x)=5*2^x. Choice B correctly constructs the exponential function with initial value 5 and base 2 from the table. A linear distractor like choice A ignores the multiplying pattern—always check ratios if differences aren't constant! 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

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

  1. P(t)=210(1.10)tP(t)=210(1.10)^t
  2. P(t)=200(1.00)t+10P(t)=200(1.00)^t+10
  3. P(t)=200(1.10)tP(t)=200(1.10)^t (correct answer)
  4. P(t)=200(0.10)tP(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 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 growth of 10% per hour starting at 200, it's exponential with a=200 and b=1.10, so P(t)=200(1.10)^t. Choice B correctly constructs the exponential function with initial value 200 and growth factor 1.10 from the description. A distractor like choice A uses decay 0.10^t instead of growth—remember growth is 1 + rate, so 1.10 for 10%. 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 15

A bacteria culture starts with 200 bacteria and grows by 10% each hour. Let B(t)B(t) be the number of bacteria after tt hours. Which exponential function models this situation?

  1. B(t)=200+0.10tB(t)=200+0.10t
  2. B(t)=200(1.10)tB(t)=200(1.10)^t (correct answer)
  3. B(t)=200(0.90)tB(t)=200(0.90)^t
  4. B(t)=210tB(t)=210^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 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 bacteria growing by 10% each hour, this is exponential growth. Starting with 200 bacteria means a = 200. Growing by 10% means each hour we multiply by 1.10 (100% + 10% = 110% = 1.10). Therefore, B(t) = 200(1.10)^t. Choice B correctly constructs B(t) = 200(1.10)^t with initial value 200 and growth factor 1.10 from the problem description. Choice A incorrectly models this as linear growth, choice C uses 0.90 which would be 10% decay not growth, and choice D omits the initial value entirely. 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 16

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

  1. f(x)=2x+3f(x)=2x+3
  2. f(x)=4x+2f(x)=4x+2
  3. f(x)=3x+4f(x)=3x+4 (correct answer)
  4. f(x)=3x+1f(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 17

An exponential function satisfies f(0)=7f(0)=7 and f(1)=14f(1)=14. Write the function in the form f(x)=a⋅bxf(x)=a\cdot b^x.

  1. f(x)=14⋅2xf(x)=14\cdot 2^x
  2. f(x)=7⋅2xf(x)=7\cdot 2^x (correct answer)
  3. f(x)=7⋅14xf(x)=7\cdot 14^x
  4. f(x)=7x+2f(x)=7x+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 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. We're given f(0) = 7 and f(1) = 14. Since f(0) = 7, we have a = 7 (because a·b⁰ = a·1 = a). To find b, we use f(1) = 14: 14 = 7·b¹, so b = 14/7 = 2. Therefore, f(x) = 7·2^x. Choice B correctly constructs f(x) = 7·2^x with initial value 7 and base 2 from the given conditions. Choice A incorrectly has initial value 14 instead of 7, which would give f(0) = 14, not 7. 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) = 7·2⁰ = 7 ✓ and f(1) = 7·2¹ = 14 ✓!

Question 18

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

  1. f(x)=4x−1f(x)=4x-1
  2. f(x)=5x+4f(x)=5x+4
  3. f(x)=4x+5f(x)=4x+5
  4. f(x)=5x−1f(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 19

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

  1. g(n)=3⋅2n−1g(n)=3\cdot 2^{n-1} (correct answer)
  2. g(n)=3⋅3n−1g(n)=3\cdot 3^{n-1}
  3. g(n)=6⋅2ng(n)=6\cdot 2^n
  4. g(n)=2n+1g(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 20

A geometric sequence has its first term a1=6a_1 = 6 and its fourth term a4=162a_4 = 162. Which function represents the nnth term of this sequence?

  1. an=6⋅27n−1a_n = 6 \cdot 27^{n-1}
  2. an=6⋅3n+1a_n = 6 \cdot 3^{n+1}
  3. an=2⋅3na_n = 2 \cdot 3^n
  4. an=6⋅3n−1a_n = 6 \cdot 3^{n-1} (correct answer)
Explanation: When you encounter a geometric sequence problem, remember that every term follows the pattern an=a1⋅rn−1a_n = a_1 \cdot r^{n-1}, where a1a_1 is the first term and rr is the common ratio. Your job is to find the common ratio using the given information. Since a1=6a_1 = 6 and a4=162a_4 = 162, you can substitute into the formula: a4=a1⋅r4−1=6⋅r3=162a_4 = a_1 \cdot r^{4-1} = 6 \cdot r^3 = 162. Solving for rr: r3=1626=27r^3 = \frac{162}{6} = 27, so r=3r = 3. Therefore, the formula is an=6⋅3n−1a_n = 6 \cdot 3^{n-1}, which is answer choice D. Let's check why the other options fail. Choice A gives an=6⋅27n−1a_n = 6 \cdot 27^{n-1}. This would make a4=6⋅273a_4 = 6 \cdot 27^3, which is astronomically large—far beyond 162. Choice B gives an=6⋅3n+1a_n = 6 \cdot 3^{n+1}. Here, a1=6⋅32=54a_1 = 6 \cdot 3^2 = 54, not 6, so this doesn't match our first term. Choice C gives an=2⋅3na_n = 2 \cdot 3^n. This makes a1=2⋅31=6a_1 = 2 \cdot 3^1 = 6 (correct first term), but a4=2⋅34=162a_4 = 2 \cdot 3^4 = 162 (seemingly correct). However, this formula gives a2=18a_2 = 18 and a3=54a_3 = 54, making the ratios inconsistent with a geometric sequence starting at 6. Always verify your geometric sequence formula by checking that it produces the correct first term and maintains a constant ratio between consecutive terms. The standard form an=a1⋅rn−1a_n = a_1 \cdot r^{n-1} is your most reliable starting point.