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. $$
\begin{aligned}
f(x) &=(x-3.5)^2-2.25.
\end{aligned}
$$ 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 $(3.5,-2.25)$ and the equation for the axis of symmetry is $x =3.5$.
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 $x$. $$
\begin{aligned}
f(0)& =(0-3.5)^2-2.25\\
&= 10\\
\textbf{y-intercept: (0,10)}
\end{aligned}
$$
Set $ y = 0$ and solve.
$$
\begin{aligned}
(x-3.5)^2-2.25 & =0\\
(x-3.5)^2&= 2.25 \\
x-3.5& = \pm\sqrt{2.25}\\
x&= 3.5\pm 1.5
\end{aligned}
$$ Find the two separate solutions: $$
\begin{array}{cl}
x= 3.5+ 1.5 &\text { or }x= 3.5- 1.5 \\
x= 5 & \text { or } x= 2\\
\textbf{x-intercept: (5,0), (2,0)}.
\end{array}
$$ Step 4: Find the domain and range of the function and sketch it.
Domain: All real numbers,
Range: $[-2.25, \infty) $.
