Question

use transformations of the graph of f(x)=5^x to graph the given function. be sure to graph and give the equation of the asymptote. use graphs to determine the functions domain and range.
g(x)= -5^x
which transformation is needed to graph the function g(x)= -5^x? choose the correct answer below.
a. the graph of f(x)=5^x should be reflected about the y-axis.
b. the graph of f(x)=5^x should be horizontally stretched by a factor of -1.
c. the graph of f(x)=5^x should be reflected about the x-axis.
d. the graph of f(x)=5^x should be vertically stretched by a factor of -1.

Explanation:

To determine the transformation from \( f(x) = 5^x \) to \( g(x) = -5^x \), we recall the rules of function transformations. A reflection about the \( x \)-axis changes \( f(x) \) to \( -f(x) \). Here, \( g(x) = -f(x) \) where \( f(x) = 5^x \), so this is a reflection about the \( x \)-axis. Option A is a reflection about the \( y \)-axis (which would be \( f(-x) \)), option B's "horizontal stretch by -1" is not a valid transformation description, and option D's "vertical stretch by -1" is incorrect as vertical stretch factors are positive and reflection about \( x \)-axis is a different transformation (multiplying by -1 reflects over \( x \)-axis, not a stretch).

Answer:

C. The graph of \( f(x) = 5^x \) should be reflected about the \( x \)-axis.