Answer
Domain: All real numbers,
Range: $[-6, \infty) $.
The graph of the parabola is shown in the figure below.
Work Step by Step
The function that we want to graph is shown below. Follow the steps outline to graph the function. $$
f(x)=5(x+1.5)^2-6.
$$ Step 1: We first determine the direction in which the function opens: The given function opens upward because the constant $a= 5$ is positive.
Step 2:Determine the vertex and the equation for the axis of symmetry. The vertex is $(-1.5,-6)$ and the equation for the axis of symmetry is $x = -1.5$.
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 $x$. $$
\begin{aligned}
f(0) & =5(0+1.5)^2-6 \\
& =5(2.25)-6 \\
& =5.25.
\end{aligned} $$ $y$-intercept: $(0,5.25)$.
Set $ y = 0$ and solve. $$
\begin{aligned}
5(x+1.5)^2 -6& =0 \\
5(x+1.5)^2 & =6 \\
(x+1.5)^2 & =\frac{6}{5} \\
(x+1.5)^2 & =1.2 \\
x+1.5 & = \pm \sqrt{1.2} \\
x & =-1.5 \pm \sqrt{1.2}.
\end{aligned}
$$ Find the two separate solutions: $$
\begin{aligned}
x & =-1.5-\sqrt{1.2} \\
& =-2.60 \\
x & =-1.5+\sqrt{1.2} \\
& =-0.40
\end{aligned}
$$ $x$-intercepts: $(-2.6,0), (-0.4, 0)$.
Step 4: Find the domain and range of the function and sketch it.
Domain: All real numbers,
Range: $[-6, \infty) $.
The graph of the parabola is shown in the figure below.
