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: Recall function transformation rules

For a function \( y = f(Bx) \), if \( B < 0 \), there is a reflection over the \( y \)-axis. If \( |B|> 1 \), the graph is horizontally compressed by a factor of \( \frac{1}{|B|} \); if \( 0<|B| < 1 \), the graph is horizontally stretched by a factor of \( \frac{1}{|B|} \). Also, for \( y = f(-Bx) \) (which is \( y=f(B(-x)) \)), the negative sign inside the function argument indicates a reflection over the \( y \)-axis.

Step2: Analyze \( g(x)=f(- 10x) \)

Given \( f(x)=x \), and \( g(x)=f(-10x) \). Here, the argument of \( f \) is \( - 10x \).

• The negative sign in \( -10x \) (comparing to \( f(x) \) where the argument is \( x \)) means the graph is reflected about the \( y \)-axis (because the transformation is of the form \( f(-kx) \) which is a reflection over \( y \)-axis).
• Now, for the horizontal transformation: the coefficient of \( x \) is \( | - 10|=10>1 \). The rule for horizontal compression/stretch: if we have \( y = f(kx) \) with \( |k|>1 \), the graph is horizontally compressed by a factor of \( \frac{1}{k} \). Here \( k = 10 \), so the graph is horizontally compressed (shrunk) by a factor of \( \frac{1}{10} \) (or we can say shrunk by a factor of 10 in terms of horizontal scaling, since compressing by \( \frac{1}{10} \) is equivalent to shrinking by a factor of 10).

Now let's analyze each option:

• Option A: Talks about reflection over \( x \)-axis (but our reflection is over \( y \)-axis) and vertical compression (but we have horizontal transformation), so A is wrong.
• Option B: Says stretched horizontally by a factor of 10, but with \( |k| = 10>1 \), it's compressed, not stretched, so B is wrong.
• Option C: Reflection about \( y \)-axis (correct, because of the negative sign) and shrunk horizontally by a factor of 10 (since \( | - 10|=10>1 \), horizontal compression by factor \( \frac{1}{10} \) or shrunk by factor 10), so C is correct.
• Option D: Reflection about \( x \)-axis (wrong, reflection is over \( y \)-axis) and stretched horizontally (wrong, it's compressed), so D is wrong.

Answer:

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