Use the exponential equation f(t)=3(0.92)^t + 7 to answer the questions.
A: What is the equation of the horizontal asymptote in this function?
B: What is the base of the power?
C: What is the initial value, f(0)?
A. f(t) = 7
B. 0.92
C. 3+7
To answer these questions using the exponential equation f(t) = 3(0.92)^t + 7, we can refer to the different parts of the equation:
A: The equation of the horizontal asymptote: An exponential function can have a horizontal asymptote if the base of the power is between -1 and 1, inclusive. In this case, the base of the power is 0.92, which is between -1 and 1. When the base is between these values, as t approaches positive or negative infinity, the function will approach a certain value known as the horizontal asymptote. In this case, as t approaches positive or negative infinity, the function approaches the constant term, which is 7. Therefore, the equation of the horizontal asymptote is y = 7.
B: The base of the power: In the exponential equation f(t) = 3(0.92)^t + 7, the base of the power is 0.92. The base is the number that is raised to the power of t in an exponential function. In this case, 0.92 is the base of the power.
C: The initial value, f(0): The initial value refers to the value of the function when t = 0. To find the initial value in this case, we substitute t = 0 into the exponential equation:
f(0) = 3(0.92)^0 + 7
Since any number raised to the power of 0 is 1, the equation simplifies to:
f(0) = 3(1) + 7 = 3 + 7 = 10
Therefore, the initial value, f(0), is 10.