Answer
The graph is shown below.
Range $\left[-\frac{49}{4},\infty\right)$.
Work Step by Step
The given function is a quadratic function:
$f(x)=x^2+3x-10$
The standard form of the quadratic function is
$f(x)=ax^2+bx+c$
Compare both equations $a=1,b=3$ and $c=-10$.
Step 1:- Parabola opens.
$a>0$, the parabola open upward.
Step 2:- Vertex.
$x-$coordinate of the vertex is $x=-\frac{b}{2a}=-\frac{3}{2(1)}=-\frac{3}{2}$.
Substitute the value of $x$ into the function.
$\Rightarrow f(-\frac{3}{2})=(-\frac{3}{2})^2+3(-\frac{3}{2})-10$
Simplify.
$\Rightarrow f(-\frac{3}{2})=\frac{9}{4}-\frac{9}{2}-10$
The LCD is $4$.
Multiply the numerator and the denominator to form LCD at the denominator.
$\Rightarrow f(-\frac{3}{2})=\frac{9}{4}-\frac{18}{4}-\frac{40}{4}$
Add all numerators.
$\Rightarrow f(-\frac{3}{2})=\frac{9-18-40}{4}$
Simplify.
$\Rightarrow f(-\frac{3}{2})=-\frac{49}{4}$
The vertex is $(-\frac{3}{2},-\frac{49}{4})$.
Step 3:- $x-$intercepts.
Replace $f(x)$ with $0$ into the given function.
$\Rightarrow 0=x^2+3x-10$
Use the quadratic formula, we have $a=1,b=3$ and $c=-10$.
$\Rightarrow x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$
Substitute all values.
$\Rightarrow x=\frac{-(3)\pm\sqrt{(3)^2-4(1)(-10)}}{2(1)}$
Simplify.
$\Rightarrow x=\frac{-3\pm\sqrt{9+40}}{2}$
$\Rightarrow x=\frac{-3\pm\sqrt{49}}{2}$
$\Rightarrow x=\frac{-3\pm7}{2}$
Separate the fractions.
$\Rightarrow x=\frac{-3+7}{2}$ or $ \frac{-3-7}{2}$
Simplify.
$\Rightarrow x=\frac{4}{2}$ or $ x=\frac{-10}{2}$
$\Rightarrow x=2$ or $ x=-5$
The $x-$intercepts are $2$ and $-5$. The parabola passes through $(2,0)$ and $(-5,0)$.
Step 4:- $y-$intercept.
Replace $x$ with $0$ in the given function.
$\Rightarrow f(0)=(0)^2+3(0)-10$
Simplify.
$\Rightarrow f(0)=-10$
The $y-$intercept is $-10$. The parabola passes through $(0,-10)$.
Step 5:- Graph.
Use the points:- vertex, $x-$intercepts and $y-$intercept to draw a parabola.
The axis of symmetry is $x=-\frac{3}{2}$.
$A=(-\frac{3}{2},-\frac{49}{4})$
$B=(2,0)$
$C=(-5,0)$
$D=(0,-10)$.
From the graph the range of the function is
$\left[-\frac{49}{4},\infty\right)$.
