To analyze the function \(y = 16(0.5)^{x}\) and determine which statement is NOT true, let's evaluate each statement:
a. The y-intercept is (0, 16):
To find the y-intercept, we set \(x = 0\):
\[
y = 16(0.5)^{0} = 16(1) = 16.
\]
So the y-intercept is indeed \((0, 16)\). This statement is true.
b. The graph is decreasing for all values of x:
The base of the exponential function \(0.5\) is less than 1. Therefore, as \(x\) increases, the value of \((0.5)^{x}\) decreases, which means \(y\) will also decrease. Thus, the graph is decreasing for all values of \(x\). This statement is true.
c. The x-intercept is (0.5, 0):
To find the x-intercept, we set \(y = 0\):
\[
16(0.5)^{x} = 0.
\]
This equation has no solution because the expression \(16(0.5)^{x}\) is never equal to zero (as an exponential function does not intersect the x-axis). Therefore, the x-intercept does not exist, and particularly it is not \((0.5, 0)\). This statement is NOT true.
d. The graph has a horizontal asymptote of y = 0:
As \(x\) approaches infinity, \(y = 16(0.5)^{x}\) approaches 0. Therefore, the graph does have a horizontal asymptote at \(y = 0\). This statement is true.
Based on the analysis, the statement that is NOT true is:
c. the x-intercept is (0.5, 0).
1 answer