Which statement about the graph of `y=16\left(0.5\right)^{x}`is NOT true?

a. the y intercept is (0, 16)
b. the graph is decreasing for all values of x.
c. the x-intercept is (0.5, 0)
d. the graph has a horizontal asymptote of y = 0

1 answer

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).

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