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

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

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

Answer 1

Let's evaluate each statement about the function \( a(t) = 500(1.07)^t \):

1. **Statement #1:** As \( t \) increases, \( a \) increases slowly at first, then quickly.
- This is true. Exponential growth starts slowly for small \( t \) values and then accelerates 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. Vertical asymptotes occur typically in rational functions or other cases, but here \( a(t) \) is defined at \( t = 0 \) and does not approach infinity as \( t \) approaches 0.

4. **Statement #4:** As \( t \) increases, \( a \) increases at a decreasing rate.
- This is false. In exponential growth, the rate of increase is always increasing, not decreasing.

5. **Statement #5:** The t-values include all real numbers greater than or equal to 0.
- This is true. The function is defined for all \( t \geq 0 \).

6. **Statement #6:** The graph increases without bound as \( t \) approaches positive infinity.
- This is true. As \( t \) increases, \( a(t) \) grows indefinitely.

Based on the analysis:

- True statements are 1, 2, 5, and 6.

Therefore, the correct response is:

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