Question

what is true about the function $h(x) = x^2 + 20x - 17$? choose two correct answers.
to graph the function $h$, shift the graph of $f(x) = x^2$ left 10 units and down 117 units.
the vertex of $h$ is $(-10, -117)$.
the axis of symmetry of function $h$ is $x = 20$.

Explanation:

Step1: Complete the square

$$\begin{align*} h(x)&=x^2+20x-17\\ &=x^2+20x+100-100-17\\ &=(x+10)^2-117 \end{align*}$$

Step2: Analyze vertex form

The vertex form is $h(x)=(x-h)^2+k$, where vertex is $(h,k)$, so vertex is $(-10,-117)$.

Step3: Analyze graph shifts

Compare to $f(x)=x^2$, $h(x)=(x+10)^2-117$ is left 10, down 117.

Step4: Check axis of symmetry

Axis of symmetry is $x=-10$, not $x=20$.

Answer:

To graph the function $h$, shift the graph of $f(x) = x^2$ left 10 units and down 117 units.
The vertex of $h$ is $(-10, -117)$.