Question

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.

(1 point)
Responses

The true statements are 2, 4, and 5.
The true statements are 2, 4, and 5.

The true statements are 1, 2, 3, and 5.
The true statements are 1, 2, 3, and 5.

The true statements are 1, 2, 5, and 6.
The true statements are 1, 2, 5, and 6.

The true statements are 2, 4, 5, and 6.
The true statements are 2, 4, 5, and 6.
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Answer 1

To evaluate the statements regarding the function \( a(t) = 500(1.07)^t \), let's analyze each statement:

1. Statement #1: As \( t \) increases, \( a \) increases slowly at first, then quickly.

   • This is true. The function is exponential, meaning it grows slowly for small values of \( t \) and more rapidly as \( t \) increases.

2. Statement #2: The y-intercept is greater than 0.

   • This is true. When \( t = 0 \), \( a(0) = 500(1.07)^0 = 500 \), which is greater than 0.

3. Statement #3: There is a vertical asymptote at \( t = 0 \).

   • This is false. The function does not have a vertical asymptote; it is defined for \( t = 0 \).

4. Statement #4: As \( t \) increases, \( a \) increases at a decreasing rate.

   • This is false. The function is increasing at an increasing rate; it grows exponentially.

5. Statement #5: The t-values include all real numbers greater than or equal to 0.

   • This is true. The function is defined for \( t \geq 0 \).

6. Statement #6: The graph increases without bound as \( t \) approaches positive infinity.

   • This is true. As \( t \) increases, \( a(t) \) will grow larger and larger without bound.

Based on this analysis, the true statements are 1, 2, 5, and 6.

So, the answer to the question is: The true statements are 1, 2, 5, and 6.