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!

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 (1,5), substitute: 5=3*1 + b gives b=2, so f(x)=3x+2. Choice A correctly constructs the linear function with m=3 and b=2 from the slope and point. A distractor like choice B uses b=5, but that would be f(1)=3+5=8≠5—always solve for b after plugging in the point. 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. 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. 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 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!

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 ✓!

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: $b = y_2 / y_1$ when x increases by 1). Then write $f(x) = a \cdot b^x$. To construct a linear function from two points, find the slope $m = (y_2 - y_1) / (x_2 - x_1)$, 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 points (0,6) and (2,54), a = 6 from x=0, then $6 b^2 = 54$ gives $b^2 = 9$, so b=3 (positive for growth), yielding $f(x)=6 \cdot 3^x$. Choice B correctly constructs the exponential function with initial value 6 and base 3 from the points. A distractor like choice A might use the wrong base by miscalculating the exponent—remember to solve for b by dividing $f(x_2) / f(x_1) = b^(x_2 - x_1)$ properly! 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 $f(x) = a \cdot b^x$, (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 $f(x) = 100 \cdot(1.1)^x$. 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. 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, differences are constantly -3, so linear with m=-3; f(0)=18 is b, giving f(x)=-3x+18 or equivalently 18-3x. Choice B correctly constructs the linear function with slope -3 and y-intercept 18 from the table. An exponential distractor like choice A might assume decay but ratios aren't constant (15/18=5/6, 12/15=4/5)—check both tests! 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. 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!

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!