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.