the graph above is a transformation of the function $x^{2}$. write an equation for the function graphed…

Explanation:

Step1: Identify vertex

The vertex of the parabola is at $(2, - 2)$. For a parabola of the form $y=a(x - h)^2+k$, where $(h,k)$ is the vertex, here $h = 2$ and $k=-2$. So the equation starts as $y=a(x - 2)^2-2$.

Step2: Find the value of $a$

The parabola passes through the point $(0,1)$. Substitute $x = 0$ and $y = 1$ into $y=a(x - 2)^2-2$. We get $1=a(0 - 2)^2-2$. Simplify the right - hand side: $1=a\times4-2$. Add 2 to both sides: $1 + 2=4a$, so $3 = 4a$. Then $a=\frac{3}{4}$.

Answer:

$g(x)=\frac{3}{4}(x - 2)^2-2$