Question

graph the function f(x)=x^{2}+8x + 15 by starting with the graph of y=x^{2} and using transformations (shifting, stretching/compressing, and/or reflecting). use the graphing tool to graph the function. click to enlarge graph

Explanation:

Step1: Rewrite the function in vertex - form

Complete the square for $f(x)=x^{2}+8x + 15$.
\[

$$\begin{align*} f(x)&=x^{2}+8x+15\\ &=(x^{2}+8x)+15\\ &=(x^{2}+8x + 16-16)+15\\ &=(x + 4)^{2}-1 \end{align*}$$

\]

Step2: Identify the transformations

The parent function is $y = x^{2}$.
For $y=(x + 4)^{2}-1$, compared to $y = x^{2}$, the graph of $y=x^{2}$ is shifted 4 units to the left (because of $x+4$) and 1 unit down (because of - 1).

Step3: Graph the function

First, graph the parent - function $y = x^{2}$, which is a parabola with vertex at the origin $(0,0)$. Then, shift the entire parabola 4 units to the left and 1 unit down. The vertex of the new parabola $y=(x + 4)^{2}-1$ is at $(-4,-1)$.

Answer:

Graph the parabola $y=(x + 4)^{2}-1$ with vertex at $(-4,-1)$ which is obtained by shifting the parabola $y = x^{2}$ 4 units to the left and 1 unit down.