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 the graph 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. graph the function g(x) = -5^x and its asymptote. graph the asymptote as a dashed line. use the graphing tool to graph the function. click to enlarge graph

Explanation:

Step1: Recall reflection rules

For a function \( y = f(x) \), reflecting over the \( x \)-axis gives \( y=-f(x) \), over \( y \)-axis gives \( y = f(-x) \). Horizontal stretch: \( y = f(kx) \), vertical stretch: \( y=kf(x) \).

Step2: Analyze \( g(x)=-5^x \) vs \( f(x)=5^x \)

Here, \( g(x)=-f(x) \) (since \( f(x)=5^x \), so \( -f(x)=-5^x \)). The transformation \( y=-f(x) \) is a reflection about the \( x \)-axis. Option A is reflection over \( y \)-axis (\( f(-x) \)), B is horizontal stretch (\( f(-x) \) is not stretch), D is vertical stretch (\( -1 \) factor vertical stretch would be \( -f(x) \)? Wait, no: vertical stretch by factor \( a \) is \( a f(x) \). Here \( a=-1 \), so \( y=-1\times f(x) \), which is reflection over \( x \)-axis (since multiplying by -1 reflects over \( x \)-axis). Wait, option C says "reflected about the \( x \)-axis", option D says "vertically stretched by a factor of -1". But reflection over \( x \)-axis is equivalent to vertical stretch by factor -1? Wait, no: vertical stretch by factor \( a \) is \( y = a f(x) \). If \( a=-1 \), it's \( y=-f(x) \), which is reflection over \( x \)-axis. But the options: let's check the options again.

Wait, the function \( g(x) = -5^x = -f(x) \), where \( f(x)=5^x \). The transformation from \( f(x) \) to \( -f(x) \) is a reflection about the \( x \)-axis. So option C: "The graph of \( f(x)=5^x \) should be reflected about the \( x \)-axis." Option D: "vertically stretched by a factor of -1" – but vertical stretch factor is usually positive, and reflection over \( x \)-axis is a type of vertical reflection, which can be considered as vertical stretch by -1, but the standard term is reflection over \( x \)-axis. So the correct transformation is reflection over \( x \)-axis, which is option C.

Answer:

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