Question
Skyla earned $500 at her part-time job as a gymnastics coach. She decided to invest the money in a certificate of deposit that earns 7 percent interest per year. The situation can be modeled by the function a(t)=500(1.07)t, where a represents the amount of money in the investment after t years. Which of the following statements about the key features of the exponential graph for this situation are true?
Statement #1: As t increases, a increases slowly at first, then quickly.
Statement #2: The y-intercept is greater than 0.
Statement #3: There is a vertical asymptote at t=0.
Statement #4: As t increases, a increases at a decreasing rate.
Statement #5: The t-values include all real numbers greater than or equal to 0.
Statement #6: The graph increases without bound as t approaches positive infinity.
Statement #1: As t increases, a increases slowly at first, then quickly.
Statement #2: The y-intercept is greater than 0.
Statement #3: There is a vertical asymptote at t=0.
Statement #4: As t increases, a increases at a decreasing rate.
Statement #5: The t-values include all real numbers greater than or equal to 0.
Statement #6: The graph increases without bound as t approaches positive infinity.
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Answered by
GPT-4o mini
Let's evaluate each statement about the function \( a(t) = 500(1.07)^t \) to determine their validity:
**Statement #1:** As t increases, a increases slowly at first, then quickly.
**True**. The function is exponential, and exponential functions start increasing slowly and gradually increase at a faster rate as \( t \) increases.
**Statement #2:** The y-intercept is greater than 0.
**True**. The y-intercept can be found by evaluating \( a(0) \). When \( t = 0 \), \( a(0) = 500(1.07)^0 = 500 \), which is greater than 0.
**Statement #3:** There is a vertical asymptote at t=0.
**False**. An exponential function like this one does not have a vertical asymptote. The graph exists at \( t = 0 \) (with value 500), and it approaches the horizontal axis (y=0) as \( t \) approaches negative infinity, but there is no vertical asymptote.
**Statement #4:** As t increases, a increases at a decreasing rate.
**False**. In an exponential function, as \( t \) increases, \( a(t) \) increases at an increasing rate. The growth is not decreasing, it actually accelerates.
**Statement #5:** The t-values include all real numbers greater than or equal to 0.
**True**. In this context, \( t \) represents time in years and starts from 0 and goes to positive infinity, so the valid domain is \( [0, \infty) \).
**Statement #6:** The graph increases without bound as t approaches positive infinity.
**True**. As \( t \) approaches positive infinity, \( a(t) = 500(1.07)^t \) grows infinitely large, and therefore the function increases without bound.
**Summary of Statements:**
- Statement #1: True
- Statement #2: True
- Statement #3: False
- Statement #4: False
- Statement #5: True
- Statement #6: True
Thus, the true statements are #1, #2, #5, and #6.
**Statement #1:** As t increases, a increases slowly at first, then quickly.
**True**. The function is exponential, and exponential functions start increasing slowly and gradually increase at a faster rate as \( t \) increases.
**Statement #2:** The y-intercept is greater than 0.
**True**. The y-intercept can be found by evaluating \( a(0) \). When \( t = 0 \), \( a(0) = 500(1.07)^0 = 500 \), which is greater than 0.
**Statement #3:** There is a vertical asymptote at t=0.
**False**. An exponential function like this one does not have a vertical asymptote. The graph exists at \( t = 0 \) (with value 500), and it approaches the horizontal axis (y=0) as \( t \) approaches negative infinity, but there is no vertical asymptote.
**Statement #4:** As t increases, a increases at a decreasing rate.
**False**. In an exponential function, as \( t \) increases, \( a(t) \) increases at an increasing rate. The growth is not decreasing, it actually accelerates.
**Statement #5:** The t-values include all real numbers greater than or equal to 0.
**True**. In this context, \( t \) represents time in years and starts from 0 and goes to positive infinity, so the valid domain is \( [0, \infty) \).
**Statement #6:** The graph increases without bound as t approaches positive infinity.
**True**. As \( t \) approaches positive infinity, \( a(t) = 500(1.07)^t \) grows infinitely large, and therefore the function increases without bound.
**Summary of Statements:**
- Statement #1: True
- Statement #2: True
- Statement #3: False
- Statement #4: False
- Statement #5: True
- Statement #6: True
Thus, the true statements are #1, #2, #5, and #6.
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