Answer
See graph
Work Step by Step
The function that we want to graph is shown below. Follow the steps outline to graph the function. $$
\begin{aligned}
f(x) &=4(x+0.625)^2-21.5625.
\end{aligned}
$$ Step 1: We first determine the direction in which the function opens: The given function opens upward because the constant $a= 4$ is positive.
Step 2: Determine the vertex and the equation for the axis of symmetry. The vertex is $(-0.625,-21.5625)$ and the equation for the axis of symmetry is $x =-0.625$.
Step 3: Find the $y$ and $x$ intercepts. To find the $y$ intercept, set $x= 0$ to find the value of $y$. Similarly, to find the $x$ intercept, set $ y= 0$ to find the values of $x$. $$
\begin{aligned}
f(0)& =4(0+0.625)^2-21.5625\\
&= -20\\
&\text{y-intercept: (0,-20)}.
\end{aligned}
$$ Set $ y = 0$ and solve. $$
\begin{aligned}
4(x+0.625)^2-21.5625& =0\\
(x+0.625)^2&= \frac{21.5625}{4} \\
x+0.625& = \pm\sqrt{5.390625}\\
x&= -0.625\pm \sqrt{5.390625}.
\end{aligned}
$$ Find the two separate solutions: $$
\begin{aligned}
x&= -0.625+ \sqrt{5.390625}\\
&\approx 1.70 \\
\text{or}\\
x&= -0.625- \sqrt{5.390625}\\
& \approx -2.95
\end{aligned}
$$ $x$-intercepts: $(-2.95,0), (1.70, 0)$.
Step 4: Find the domain and range of the function and sketch it.
Domain: All real numbers,
Range: $[-21.5625, \infty) $.
