Questions: Describe a function g(x) in terms of f(x) if the graph of g is obtained by reflecting the graph of f about the x-axis and if it is horizontally stretched by a factor of 2 when compared to the graph of f. g(x)=A f(B x)+C where A= B= C=

Solution Steps

To describe the function \( g(x) \) in terms of \( f(x) \), we need to apply two transformations to \( f(x) \): a reflection about the x-axis and a horizontal stretch by a factor of 2. Reflecting a function about the x-axis involves multiplying the function by -1. A horizontal stretch by a factor of 2 involves replacing \( x \) with \( \frac{x}{2} \) in the function. Therefore, the transformed function \( g(x) \) can be expressed as \( g(x) = -f\left(\frac{x}{2}\right) \).

We start with the function \( f(x) = x^2 \).

To obtain the function \( g(x) \), we need to reflect \( f(x) \) about the x-axis and horizontally stretch it by a factor of 2. The reflection about the x-axis is achieved by multiplying \( f(x) \) by -1, and the horizontal stretch is achieved by replacing \( x \) with \( \frac{x}{2} \). Thus, we have: \[ g(x) = -f\left(\frac{x}{2}\right) \]

Substituting \( x = 4 \) into the expression for \( g(x) \): \[ g(4) = -f\left(\frac{4}{2}\right) = -f(2) \] Now, we calculate \( f(2) \): \[ f(2) = 2^2 = 4 \] Therefore, \[ g(4) = -4 \]

Final Answer