Question

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Explanation:

Step1: Recall Transformations of Quadratic Functions

The parent function is \( f(x) = x^2 \). The given function is \( g(x)=4(x - 1)^2-5 \). For quadratic functions of the form \( y = a(x - h)^2 + k \), the transformations from \( y=x^2 \) are: a vertical stretch (if \( |a|>1 \)) or compression (if \( |a|<1 \)) by a factor of \( |a| \), a horizontal shift of \( h \) units (right if \( h>0 \), left if \( h<0 \)), and a vertical shift of \( k \) units (up if \( k>0 \), down if \( k<0 \)).

Step2: Analyze Vertical Stretch/Compression

Here, \( a = 4 \), since \( |4|>1 \), the graph of \( g(x) \) is a vertical stretch of the graph of \( f(x)=x^2 \) by a factor of 4.

Step3: Analyze Horizontal Shift

The term \( (x - 1) \) means \( h = 1 \), so the graph is shifted 1 unit to the right.

Step4: Analyze Vertical Shift

The term \( - 5 \) means \( k=-5 \), so the graph is shifted 5 units down.

Answer:

The graph of \( r \) (should be \( g \)) is a \(\boldsymbol{\text{vertical stretch}}\) by a factor of 4, and a translation \(\boldsymbol{\text{1 unit right}}\) and \(\boldsymbol{\text{5 units down}}\) of the graph of \( f \).