The exponential function is determined to be y = 5 · 2^x by analyzing the given table, where a = 5 and b = 2. To predict the value when x = 5, the function computes a value of 160. As x increases, the function exhibits exponential growth.
Step-by-step explanation:
Determining the Exponential Function Constants
To find the values of a and b in the exponential function y = a · b^x, we look at the table of x and y values:
When x = 0, y = 5. This tells us that a = 5, since b^0 is always 1.
Next, when x = 1, y = 10. Using the previously found value of a, we get 10 = 5 · b^1, which simplifies to b = 2.
Therefore, the exponential function is y = 5 · 2^x.
Predicting Future Values
To predict the value of y when x = 5, we use the function: y = 5 · 2^5 which equals 160. Thus, the predicted value is 160.
Behavior of the Exponential Function
As x increases, the function y = 5 · 2^x shows exponential growth. Each increase in x results in the doubling of y, since the base b is greater than 1. This behavior reflects the compound growth characteristic of exponential functions.
The probable question can be:
"Consider the exponential function \(y = a \cdot b^x\), where \(a\) and \(b\) are constants. Using the given table of \(x\) and \(y\) values:
x & y
0 & 5
1 & 10
2 & 20
3 & 40
4 & 80
(a) Determine the values of \(a\) and \(b\) for the exponential function.
(b) Use the exponential function to predict the value of \(y\) when \(x = 5\).
(c) Discuss the behavior of the function as \(x\) increases.
Ensure to show your calculations and provide a clear explanation for each part."