Question

describe the transformations of the parent function $f(x)=x^{2}$ represented by $g(x)=2(x + 1)^{2}-8$.

Explanation:

Step1: Analyze horizontal shift

The function $g(x)=2(x + 1)^2-8$ has $x+1$ inside the square - function. For a parent function $y = f(x)$ and a transformed function $y=f(x + h)$, if $h>0$, the graph shifts left by $h$ units. Here $h = 1$, so the graph of $y=x^2$ shifts left 1 unit.

Step2: Analyze vertical stretch

The coefficient of $(x + 1)^2$ is 2. For a parent function $y = f(x)$ and a transformed function $y=af(x)$ where $a>1$, the graph of $y = f(x)$ is vertically stretched by a factor of $a$. So the graph of $y=x^2$ is vertically stretched by a factor of 2.

Step3: Analyze vertical shift

The constant term in $g(x)$ is - 8. For a parent function $y = f(x)$ and a transformed function $y=f(x)+k$, if $k<0$, the graph shifts down by $|k|$ units. Here $k=-8$, so the graph of $y=x^2$ shifts down 8 units.

Answer:

The parent - function $f(x)=x^2$ is shifted left 1 unit, vertically stretched by a factor of 2, and shifted down 8 units to get $g(x)=2(x + 1)^2-8$.