Question

suppose that ( f(x) = x^2 ) and ( g(x) = -\frac{1}{4}(x + 7)^2 ). which statement best compares the graph of ( g(x) ) with the graph of ( f(x) )?
a. the graph of ( g(x) ) is the graph of ( f(x) ) compressed vertically, flipped over the ( x )-axis, and shifted 7 units to the left
b. the graph of ( g(x) ) is the graph of ( f(x) ) stretched vertically, flipped over the ( x )-axis, and shifted 7 units to the right
c. the graph of ( g(x) ) is the graph of ( f(x) ) compressed vertically, flipped over the ( x )-axis, and shifted 7 units to the right
d. the graph of ( g(x) ) is the graph of ( f(x) ) stretched vertically, flipped over the ( x )-axis, and shifted 7 units to the left

Explanation:

Step1: Analyze vertical transformation

For a function \( y = a f(x) \), if \( |a| < 1 \), it's a vertical compression; if \( a < 0 \), it's a reflection over the x - axis. Here, \( F(x)=x^{2} \) and \( G(x)=-\frac{1}{4}(x + 7)^{2} \), \( a=-\frac{1}{4} \). Since \( |-\frac{1}{4}|=\frac{1}{4}<1 \), there is a vertical compression, and the negative sign means a reflection (flip) over the x - axis.

Step2: Analyze horizontal transformation

For a function \( y = f(x + h) \), if \( h>0 \), the graph shifts \( h \) units to the left; if \( h < 0 \), it shifts \( |h| \) units to the right. In \( G(x)=-\frac{1}{4}(x + 7)^{2} \), we can think of it as \( G(x)=-\frac{1}{4}F(x + 7) \) (since \( F(x)=x^{2} \), so \( F(x + 7)=(x + 7)^{2} \)). Here, \( h = 7>0 \), so the graph shifts 7 units to the left.

Answer:

A. The graph of \( G(x) \) is the graph of \( F(x) \) compressed vertically, flipped over the \( x \) - axis, and shifted 7 units to the left.