Question

which statement best compares the graphs of the following functions $f(x)$ and $g(x)$?
$f(x)=x^2$
$g(x)=3x^2 - 8$
show your work here
\bigcirc compared to $f(x)$, the graph of $g(x)$ is more vertically stretched and is shifted 8 units down.
\bigcirc compared to $f(x)$, the graph of $g(x)$ is more vertically compressed and is shifted 8 units down.
\bigcirc compared to $f(x)$, the graph of $g(x)$ is more vertically stretched and is shifted 8 units up.
\bigcirc compared to $f(x)$, the graph of $g(x)$ is more vertically compressed and is shifted 8 units up.

Explanation:

Step1: Analyze vertical stretch factor

For a function transformation $g(x) = a \cdot f(x) + k$, when $|a|>1$, the graph is vertically stretched. Here, $f(x)=x^2$, $g(x)=3x^2-8$, so $a=3>1$. This means $g(x)$ is vertically stretched compared to $f(x)$.

Step2: Analyze vertical shift

In $g(x) = a \cdot f(x) + k$, $k$ determines vertical shift. If $k<0$, the graph shifts down. Here, $k=-8$, so the graph shifts 8 units down.

Answer:

Compared to $f(x)$, the graph of $g(x)$ is more vertically stretched and is shifted 8 units down.