Question

select the correct answer. each statement describes a transformation of the graph of f(x) = x. which statement correctly describes the graph of g(x) if g(x) = f(-10x)? a. it is the graph of f(x) reflected about the x - axis and shrunk or compressed vertically by a factor of 10. b. it is the graph of f(x) reflected about the y - axis and stretched horizontally by a factor of 10. c. it is the graph of f(x) reflected about the y - axis and shrunk or compressed horizontally by a factor of 10. d. it is the graph of f(x) reflected about the x - axis and stretched horizontally by a factor of 10.

Explanation:

Step1: Analyze the transformation of \( g(x) = f(-10x) \) from \( f(x)=x \)

First, recall the rules of function transformations:

• For a function \( y = f(-x) \), it is a reflection of \( y = f(x) \) about the \( y \)-axis.
• For a function \( y = f(kx) \) where \( |k|> 1 \), it is a horizontal compression (shrink) of \( y = f(x) \) by a factor of \( \frac{1}{|k|} \); if \( 0<|k|<1 \), it is a horizontal stretch.

Given \( f(x)=x \), then \( g(x)=f(- 10x)=- 10x \) (since \( f(t) = t \), substitute \( t=-10x \)). But we analyze the transformation step - by - step from \( f(x) \) to \( f(-10x) \):

1. First, consider the transformation from \( f(x) \) to \( f(-x) \): This is a reflection about the \( y \)-axis.
2. Then, consider the transformation from \( f(-x) \) to \( f(-10x) \): Let \( u=-x \), then \( f(-10x)=f(10u) \) (where \( u = - x \)). For the function \( y = f(kx) \) with \( k = 10>1 \), the graph of \( y = f(x) \) is horizontally compressed by a factor of \( \frac{1}{10} \) (or shrunk horizontally by a factor of \( \frac{1}{10} \), which is equivalent to saying it is shrunk horizontally by a factor of 10 in terms of the transformation of the input variable). Wait, actually, the rule is: if we have \( y = f(kx) \), the horizontal scaling factor is \( \frac{1}{k} \). So when \( k = 10 \), the graph of \( f(x) \) is horizontally compressed (shrunk) by a factor of \( \frac{1}{10} \) (or we can say it is shrunk horizontally by a factor of 10 in the sense that the input is multiplied by 10, so the graph is compressed towards the \( y \)-axis). And since we first had a reflection about the \( y \)-axis (from \( f(x) \) to \( f(-x) \)) and then a horizontal compression by a factor of \( \frac{1}{10} \) (or shrunk horizontally by a factor of 10), let's check the options:

Option B: "It is the graph of \( f(x) \) reflected about the \( y \)-axis and stretched horizontally by a factor of 10." - Stretching horizontally by a factor of 10 would correspond to \( k=\frac{1}{10} \), but we have \( k = 10 \), so it's a compression, not a stretch. Eliminate B.

Option D: "It is the graph of \( f(x) \) reflected about the \( x \)-axis and stretched horizontally by a factor of 10." - There is no reflection about the \( x \)-axis (reflection about \( x \)-axis is \( y=-f(x) \)), and also the horizontal transformation is wrong. Eliminate D.

Option A: "It is the graph of \( f(x) \) reflected about the \( x \)-axis and shrunk or compressed vertically by a factor of 10." - There is no reflection about the \( x \)-axis (since \( g(x)=f(-10x) \), and \( f(x)=x \), \( g(x)=-10x \), but the reflection is about the \( y \)-axis, not \( x \)-axis, and the transformation is horizontal, not vertical). Eliminate A.

Option C: "It is the graph of \( f(x) \) reflected about the \( y \)-axis and shrunk or compressed horizontally by a factor of 10." - First, \( f(-x) \) is a reflection about the \( y \)-axis. Then \( f(-10x) \) is \( f(-x) \) with \( x \) replaced by \( 10x \), so the graph is horizontally compressed (shrunk) by a factor of 10 (since \( k = 10>1 \), the horizontal scaling factor is \( \frac{1}{10} \), which means the graph is compressed towards the \( y \)-axis, i.e., shrunk horizontally by a factor of 10).

Answer:

C. It is the graph of \( f(x) \) reflected about the \( y \)-axis and shrunk or compressed horizontally by a factor of 10.