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.$$
\begin{aligned}
h(x)=(x+2)^2-5.
\end{aligned}
$$ Step 1: We first determine the direction in which the function opens: The given function opens upward because the constant, $a= 1$ is positive.
Step 2:Determine the vertex and the equation for the axis of symmetry. The vertex is $(-2,-5)$ and the equation for the axis of symmetry is $x = -2$
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 $x$.
$$
\begin{aligned}
h(0)& =(0+2)^2-5= -1\\
&\textbf{y-intercept: (0,-1)}.
\end{aligned}
$$ Set $ y = 0$ and solve. $$
\begin{aligned}
(x+2)^2-5& =0\\
(x+2)^2&= 5 \\
x+2& =\pm\sqrt{5}\\
x&= -2\pm \sqrt{5}.
\end{aligned}
$$ Find the two separate solutions: $$
\begin{aligned}
x&=-2+\sqrt{5}\approx = 0.24 \\
\textbf{or}\\
x&= -2-\sqrt{5}\approx = -4.24.
\end{aligned}
$$ $x$-intercepts: $(-4.24,0), (0.24 , 0)$.
Step 4: Find the domain and range of the function and sketch it.
Domain: All real numbers,
Range: $[-5, \infty) $.
