Question

describe the transformation of $f(x) = x^2$ represented by $g(x) = (x + 3)^2 - 5$. then identify the graph of each function.
the graph of $g$ is a translation __________ and __________ of the graph of $f$.
::: 3 units left
::: 3 units right
::: 3 units up
::: 3 units down
::: 5 units up
::: 5 units down
::: 5 units left
::: 5 units right

Explanation:

Step1: Analyze horizontal translation

For a function transformation of the form $g(x) = f(x + h)$, when $h>0$, the graph shifts left by $h$ units. Here, $g(x)=(x+3)^2-5$ and $f(x)=x^2$, so $h=3$, meaning a shift 3 units left.

Step2: Analyze vertical translation

For a function transformation of the form $g(x) = f(x) - k$, when $k>0$, the graph shifts down by $k$ units. Here, $g(x)=(x+3)^2-5 = f(x+3)-5$, so $k=5$, meaning a shift 5 units down.

Step3: Match to correct graph

The graph with $f(x)=x^2$ (vertex at $(0,0)$) and $g(x)$ with vertex at $(-3,-5)$ (shifted 3 left, 5 down) is the bottom-left graph.

Answer:

The graph of $g$ is a translation 3 units left and 5 units down of the graph of $f$.
The correct corresponding graph is the bottom-left grid (with $f$ opening from $(0,0)$ and $g$ opening from $(5,3)$ on the x-y axes, shifted left 3 and down 5).