Question 1 A) Use the Table to answer the question (1 point) Miguel is planning to invest $5,000. He plans to withdraw his investment and earnings in approximately 10 years. With Option #1, his money will grow according to the function p(x)=5,000(1.06)xp(x)=5,000(1.06)x. With Option #2 his money will grow according to the function q(x)=500x+5,000q(x)=500x+5,000. He creates sequences for both functions. Which option has the greater rate of change between years 9 and 10? Which option should he choose for his money? Option # has the greater rate of change between years 9 and 10. If Miguel plans to invest his money for only 10 years, he should choose Option # Question 2 A)Liam is using sequences to compare the growth rates of h(x)=1.2x and j(x) 1.2xh(x)=1.2x and j(x) 1.2x. Which statement correctly describes how Liam should do this and what he will observe?(1 point) Responses Liam should look at where one sequence has terms greater than the terms in the other sequence. The growth rate of j(x)=1.2xj(x)=1.2x is only greater than the growth rate of h(x)=1.2xh(x)=1.2x when its terms are greater. Liam should look at where one sequence has terms greater than the terms in the other sequence. The growth rate of j times x is equal to 1 point 2 to the x th power is only greater than the growth rate of h of x is equal to 1 point 2 x when its terms are greater. Liam should compare the rates of change of the terms in both sequences. The growth rate of h(x)=1.2xh(x)=1.2x will quickly surpass the growth rate of j(x)=1.2xj(x)=1.2x Liam should compare the rates of change of the terms in both sequences. The growth rate of h of x is equal to 1 point 2 x will quickly surpass the growth rate of j times x is equal to 1 point 2 to the x th power Liam should compare the rates of change of the terms in both sequences. The growth rate of j(x)=1.2xj(x)=1.2x will quickly surpass the growth rate of h(x)=1.2xh(x)=1.2xLiam should compare the rates of change of the terms in both sequences. The growth rate of j times x is equal to 1 point 2 to the x th power will quickly surpass the growth rate of h of x is equal to 1 point 2 x Liam should look at where one sequence has terms greater than the terms in the other sequence. The growth rate of h(x)=1.2xh(x)=1.2x is only greater than the growth rate of j(x)=1.2xj(x)=1.2x when its terms are greater. Liam should look at where one sequence has terms greater than the terms in the other sequence. The growth rate of h of x is equal to 1 point 2 x is only greater than the growth rate of j times x is equal to 1 point 2 to the x th power when its terms are greater. Question 3 A)Austin is using graphs to compare the growth rates of g(x)=1.3x and f(x)=1.3xg(x)=1.3x and f(x)=1.3x. Which statement correctly describes how Austin should do this and what he will observe?(1 point) Responses Austin should compare the steepness of the curves. The growth rate of f(x)=1.3xf(x)=1.3x will quickly surpass the growth rate of g(x)=1.3xg(x)=1.3x Austin should compare the steepness of the curves. The growth rate of f of x is equal to 1 point 3 to the x th power will quickly surpass the growth rate of g of x is equal to 1 point 3 x Austin should find where one curve is above the other curve on the graph. The growth rate of g(x)=1.3xg(x)=1.3x is only greater than the growth rate of f(x)=1.3xf(x)=1.3x to the right of their right-most intersection point. Austin should find where one curve is above the other curve on the graph. The growth rate of g of x is equal to 1 point 3 x is only greater than the growth rate of f of x is equal to 1 point 3 to the x th power to the right of their right-most intersection point. Austin should find where one curve is above the other curve on the graph. The growth rate of f(x)=1.3xf(x)=1.3x is only greater than the growth rate of g(x)=1.3xg(x)=1.3x to the right of their right-most intersection point. Austin should find where one curve is above the other curve on the graph. The growth rate of f of x is equal to 1 point 3 to the x th power is only greater than the growth rate of g of x is equal to 1 point 3 x to the right of their right-most intersection point. Austin should compare the steepness of the curves. The growth rate of g(x)=1.3xg(x)=1.3x will quickly surpass the growth rate of f(x)=1.3xf(x)=1.3xAustin should compare the steepness of the curves. The growth rate of g of x is equal to 1 point 3 x will quickly surpass the growth rate of f of x is equal to 1 point 3 to the x th power Question 4 A) Marshall is comparing the growth rates of f(x)=6(1.5)xf(x)=6(1.5)x and g(x)=5x+10g(x)=5x+10 using this graph. Based on the graph, he concludes that the growth rate of g(x)=5x+10g(x)=5x+10 is always greater than the growth rate of f(x)=6(1.5)xf(x)=6(1.5)x. Where is his mistake? (1 point) Responses The growth rate of g(x)=5x+10g(x)=5x+10 is only greater than the growth rate of f(x)=6(1.5)xf(x)=6(1.5)x when x is greater than approximatel -1.3 The growth rate of g of x is equal to 5 x plus 10 is only greater than the growth rate of f of x is equal to 6 times 1 point 5 to the x th power when x is greater than approximatel -1.3 Marshall is only considering a small portion of the domains of the functions. If he went farther to the right on the graph, he would observe that the growth rate of f(x)=6(1.5)xf(x)=6(1.5)x eventually surpasses the growth rate of g(x)=5x+10g(x)=5x+10 for negative values of x. Marshall is only considering a small portion of the domains of the functions. If he went farther to the right on the graph, he would observe that the growth rate of f of x is equal to 6 times 1 point 5 to the x th power eventually surpasses the growth rate of g of x is equal to 5 x plus 10 for negative values of x. Marshall is backward. The growth rate of f(x)=6(1.5)xf(x)=6(1.5)x is always greater than the growth rate of g(x)=5x+10g(x)=5x+10Marshall is backward. The growth rate of f of x is equal to 6 times 1 point 5 to the x th power is always greater than the growth rate of g of x is equal to 5 x plus 10 Marshall is only considering a small portion of the domains of the functions. If he went farther to the right on the graph, he would observe that the growth rate of f(x)=6(1.5)xf(x)=6(1.5)x eventually surpasses the growth rate of g(x)=5x+10g(x)=5x+10 for large values of x. Marshall is only considering a small portion of the domains of the functions. If he went farther to the right on the graph, he would observe that the growth rate of f of x is equal to 6 times 1 point 5 to the x th power eventually surpasses the growth rate of g of x is equal to 5 x plus 10 for large values of x.

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Let's analyze each question step by step.

Question 1:

Option # has the greater rate of change between years 9 and 10. If Miguel plans to invest his money for only 10 years, he should choose Option #1.

Explanation: We need to calculate the growth for both options at x = 9 and x = 10.

Option #1: $ p(9) = 5000(1.06)^9 $ and $ p(10) = 5000(1.06)^{10} $

Option #2: $ q(9) = 500(9) + 5000 $ and $ q(10) = 500(10) + 5000 $

Calculate and compare the values:

For Option #1, compute:

$ p(9) = 5000(1.06)^9 \approx 5000 \times 1.689 = 8445 $

$ p(10) = 5000(1.06)^{10} \approx 5000 \times 1.790 = 8950 $

Rate of change: $ 8950 - 8445 = 505 $

For Option #2, compute:

$ q(9) = 500(9) + 5000 = 4500 + 5000 = 9500 $

$ q(10) = 500(10) + 5000 = 5000 + 5000 = 10000 $

Rate of change: $ 10000 - 9500 = 500 $

Between years 9 and 10, the rate of change is greater for Option #1 (505) compared to Option #2 (500).

Question 2:

Liam should compare the rates of change of the terms in both sequences. The growth rate of h(x) = 1.2^x will quickly surpass the growth rate of j(x) = 1.2x.

Explanation: The function $ h(x) = 1.2^x $ is an exponential function and will eventually grow faster than the linear function $ j(x) = 1.2x $.

Question 3:

Austin should compare the steepness of the curves. The growth rate of f(x) = 1.3^x will quickly surpass the growth rate of g(x) = 1.3x.

Explanation: Similar to Question 2, $ f(x) = 1.3^x $ is the exponential function, which will increase at a faster rate than $ g(x) = 1.3x $, a linear function.

Question 4:

Marshall is only considering a small portion of the domains of the functions. If he went farther to the right on the graph, he would observe that the growth rate of f(x) = 6(1.5)^x eventually surpasses the growth rate of g(x) = 5x + 10 for large values of x.

Explanation: $ f(x) $ is an exponential function, and as x increases, it grows faster than the linear function $ g(x) $. Marshall's conclusion is incorrect if he believes that $ g(x) $ will always be greater.

Summary of Answers:

Option #1; he should choose Option #1.

Compare rates of change; growth of $ h(x) $ will surpass $ j(x) $.

Compare steepness; growth of $ f(x) $ will surpass $ g(x) $.

He's only considering a small portion; $ f(x) $ will surpass $ g(x) $ for large values of x.

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Question 1 A) Use the Table to answer the question (1 point) Miguel is planning to invest $5,000. He plans to withdraw his investment and earnings in approximately 10 years. With Option #1, his money will grow according to the function p(x)=5,000(1.06)xp(x)=5,000(1.06)x. With Option #2 his money will grow according to the function q(x)=500x+5,000q(x)=500x+5,000. He creates sequences for both functions. Which option has the greater rate of change between years 9 and 10? Which option should he choose for his money? Option # has the greater rate of change between years 9 and 10. If Miguel plans to invest his money for only 10 years, he should choose Option # Question 2 A)Liam is using sequences to compare the growth rates of h(x)=1.2x and j(x) 1.2xh(x)=1.2x and j(x) 1.2x. Which statement correctly describes how Liam should do this and what he will observe?(1 point) Responses Liam should look at where one sequence has terms greater than the terms in the other sequence. The growth rate of j(x)=1.2xj(x)=1.2x is only greater than the growth rate of h(x)=1.2xh(x)=1.2x when its terms are greater. Liam should look at where one sequence has terms greater than the terms in the other sequence. The growth rate of j times x is equal to 1 point 2 to the x th power is only greater than the growth rate of h of x is equal to 1 point 2 x when its terms are greater. Liam should compare the rates of change of the terms in both sequences. The growth rate of h(x)=1.2xh(x)=1.2x will quickly surpass the growth rate of j(x)=1.2xj(x)=1.2x Liam should compare the rates of change of the terms in both sequences. The growth rate of h of x is equal to 1 point 2 x will quickly surpass the growth rate of j times x is equal to 1 point 2 to the x th power Liam should compare the rates of change of the terms in both sequences. The growth rate of j(x)=1.2xj(x)=1.2x will quickly surpass the growth rate of h(x)=1.2xh(x)=1.2xLiam should compare the rates of change of the terms in both sequences. The growth rate of j times x is equal to 1 point 2 to the x th power will quickly surpass the growth rate of h of x is equal to 1 point 2 x Liam should look at where one sequence has terms greater than the terms in the other sequence. The growth rate of h(x)=1.2xh(x)=1.2x is only greater than the growth rate of j(x)=1.2xj(x)=1.2x when its terms are greater. Liam should look at where one sequence has terms greater than the terms in the other sequence. The growth rate of h of x is equal to 1 point 2 x is only greater than the growth rate of j times x is equal to 1 point 2 to the x th power when its terms are greater. Question 3 A)Austin is using graphs to compare the growth rates of g(x)=1.3x and f(x)=1.3xg(x)=1.3x and f(x)=1.3x. Which statement correctly describes how Austin should do this and what he will observe?(1 point) Responses Austin should compare the steepness of the curves. The growth rate of f(x)=1.3xf(x)=1.3x will quickly surpass the growth rate of g(x)=1.3xg(x)=1.3x Austin should compare the steepness of the curves. The growth rate of f of x is equal to 1 point 3 to the x th power will quickly surpass the growth rate of g of x is equal to 1 point 3 x Austin should find where one curve is above the other curve on the graph. The growth rate of g(x)=1.3xg(x)=1.3x is only greater than the growth rate of f(x)=1.3xf(x)=1.3x to the right of their right-most intersection point. Austin should find where one curve is above the other curve on the graph. The growth rate of g of x is equal to 1 point 3 x is only greater than the growth rate of f of x is equal to 1 point 3 to the x th power to the right of their right-most intersection point. Austin should find where one curve is above the other curve on the graph. The growth rate of f(x)=1.3xf(x)=1.3x is only greater than the growth rate of g(x)=1.3xg(x)=1.3x to the right of their right-most intersection point. Austin should find where one curve is above the other curve on the graph. The growth rate of f of x is equal to 1 point 3 to the x th power is only greater than the growth rate of g of x is equal to 1 point 3 x to the right of their right-most intersection point. Austin should compare the steepness of the curves. The growth rate of g(x)=1.3xg(x)=1.3x will quickly surpass the growth rate of f(x)=1.3xf(x)=1.3xAustin should compare the steepness of the curves. The growth rate of g of x is equal to 1 point 3 x will quickly surpass the growth rate of f of x is equal to 1 point 3 to the x th power Question 4 A) Marshall is comparing the growth rates of f(x)=6(1.5)xf(x)=6(1.5)x and g(x)=5x+10g(x)=5x+10 using this graph. Based on the graph, he concludes that the growth rate of g(x)=5x+10g(x)=5x+10 is always greater than the growth rate of f(x)=6(1.5)xf(x)=6(1.5)x. Where is his mistake? (1 point) Responses The growth rate of g(x)=5x+10g(x)=5x+10 is only greater than the growth rate of f(x)=6(1.5)xf(x)=6(1.5)x when x is greater than approximatel -1.3 The growth rate of g of x is equal to 5 x plus 10 is only greater than the growth rate of f of x is equal to 6 times 1 point 5 to the x th power when x is greater than approximatel -1.3 Marshall is only considering a small portion of the domains of the functions. If he went farther to the right on the graph, he would observe that the growth rate of f(x)=6(1.5)xf(x)=6(1.5)x eventually surpasses the growth rate of g(x)=5x+10g(x)=5x+10 for negative values of x. Marshall is only considering a small portion of the domains of the functions. If he went farther to the right on the graph, he would observe that the growth rate of f of x is equal to 6 times 1 point 5 to the x th power eventually surpasses the growth rate of g of x is equal to 5 x plus 10 for negative values of x. Marshall is backward. The growth rate of f(x)=6(1.5)xf(x)=6(1.5)x is always greater than the growth rate of g(x)=5x+10g(x)=5x+10Marshall is backward. The growth rate of f of x is equal to 6 times 1 point 5 to the x th power is always greater than the growth rate of g of x is equal to 5 x plus 10 Marshall is only considering a small portion of the domains of the functions. If he went farther to the right on the graph, he would observe that the growth rate of f(x)=6(1.5)xf(x)=6(1.5)x eventually surpasses the growth rate of g(x)=5x+10g(x)=5x+10 for large values of x. Marshall is only considering a small portion of the domains of the functions. If he went farther to the right on the graph, he would observe that the growth rate of f of x is equal to 6 times 1 point 5 to the x th power eventually surpasses the growth rate of g of x is equal to 5 x plus 10 for large values of x.

1 answer

Let's analyze each question step by step.

Question 1:

Option # has the greater rate of change between years 9 and 10. If Miguel plans to invest his money for only 10 years, he should choose Option #1.

Explanation: We need to calculate the growth for both options at x = 9 and x = 10.

Option #1: $ p(9) = 5000(1.06)^9 $ and $ p(10) = 5000(1.06)^{10} $
Option #2: $ q(9) = 500(9) + 5000 $ and $ q(10) = 500(10) + 5000 $

Calculate and compare the values:

For Option #1, compute:
- $ p(9) = 5000(1.06)^9 \approx 5000 \times 1.689 = 8445 $
- $ p(10) = 5000(1.06)^{10} \approx 5000 \times 1.790 = 8950 $
- Rate of change: $ 8950 - 8445 = 505 $
For Option #2, compute:
- $ q(9) = 500(9) + 5000 = 4500 + 5000 = 9500 $
- $ q(10) = 500(10) + 5000 = 5000 + 5000 = 10000 $
- Rate of change: $ 10000 - 9500 = 500 $

Between years 9 and 10, the rate of change is greater for Option #1 (505) compared to Option #2 (500).

Question 2:

Liam should compare the rates of change of the terms in both sequences. The growth rate of h(x) = 1.2^x will quickly surpass the growth rate of j(x) = 1.2x.

Explanation: The function $ h(x) = 1.2^x $ is an exponential function and will eventually grow faster than the linear function $ j(x) = 1.2x $.

Question 3:

Austin should compare the steepness of the curves. The growth rate of f(x) = 1.3^x will quickly surpass the growth rate of g(x) = 1.3x.

Explanation: Similar to Question 2, $ f(x) = 1.3^x $ is the exponential function, which will increase at a faster rate than $ g(x) = 1.3x $, a linear function.

Question 4:

Marshall is only considering a small portion of the domains of the functions. If he went farther to the right on the graph, he would observe that the growth rate of f(x) = 6(1.5)^x eventually surpasses the growth rate of g(x) = 5x + 10 for large values of x.

Explanation: $ f(x) $ is an exponential function, and as x increases, it grows faster than the linear function $ g(x) $. Marshall's conclusion is incorrect if he believes that $ g(x) $ will always be greater.

Summary of Answers:

Option #1; he should choose Option #1.
Compare rates of change; growth of $ h(x) $ will surpass $ j(x) $.
Compare steepness; growth of $ f(x) $ will surpass $ g(x) $.
He's only considering a small portion; $ f(x) $ will surpass $ g(x) $ for large values of x.

Ask a New Question or answer this question.

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