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!

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 ✓!

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 ✓!

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!

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!

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 ✓!

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!

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 ✓!

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 ✓)!

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!