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.
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Option #1: \( p(9) = 5000(1.06)^9 \) and \( p(10) = 5000(1.06)^{10} \)
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Option #2: \( q(9) = 500(9) + 5000 \) and \( q(10) = 500(10) + 5000 \)
Calculate and compare the values:
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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: \( 8950 - 8445 = 505 \)
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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.