Question

identify the graph of $f(x) = \frac{1}{4}(x + 10)^2$

compare the graph to the graph of $f(x) = x^2$.

the graph of $f$ is a \underline{\quad\quad} translation \underline{\quad\quad} unit(s) \underline{\quad\quad} and a \underline{\quad\quad} by a factor of \underline{\quad\quad} of the graph of $f$

options: $10$, $\frac{1}{4}$, $4$, up, down, left, right, vertical, horizontal, vertical shrink, vertical stretch, horizontal shrink, horizontal stretch

Explanation:

Step1: Recall transformation rules

For a quadratic function in the form \( f(x)=a(x - h)^2 + k \), compared to \( f(x)=x^2 \):

• Horizontal translation: If \( h>0 \), shift right \( h \) units; if \( h<0 \), shift left \( |h| \) units.
• Vertical translation: If \( k>0 \), shift up \( k \) units; if \( k<0 \), shift down \( |k| \) units.
• Vertical stretch/shrink: If \( |a|>1 \), vertical stretch; if \( 0<|a|<1 \), vertical shrink.

Given \( f(x)=\frac{1}{4}(x + 10)^2 \), here \( h=- 10 \), \( k = 0 \), \( a=\frac{1}{4} \).

Step2: Analyze horizontal translation

Since \( h=-10 \), the graph of \( f(x)=\frac{1}{4}(x + 10)^2 \) is a horizontal translation of \( f(x)=x^2 \) 10 units to the left.

Step3: Analyze vertical transformation

Since \( |a|=\frac{1}{4}<1 \), the graph of \( f(x)=\frac{1}{4}(x + 10)^2 \) is a vertical shrink of the graph of \( f(x)=x^2 \) by a factor of \( \frac{1}{4} \).

Answer:

The graph of \( f \) is a \(\boldsymbol{\text{horizontal}}\) translation \(\boldsymbol{10}\) unit(s) \(\boldsymbol{\text{left}}\) and a \(\boldsymbol{\text{vertical shrink}}\) by a factor of \(\boldsymbol{\frac{1}{4}}\) of the graph of \( f(x)=x^2 \)