Answer
The graph is shown below.
Range $\left[-\frac{81}{8},\infty\right)$.
Work Step by Step
The given function is a quadratic function:
$f(x)=2x^2-7x-4$
The standard form of the quadratic function is
$f(x)=ax^2+bx+c$
Compare both equations $a=2,b=-7$ and $c=-4$.
Step 1:- Parabola opens.
$a>0$, the parabola open upward.
Step 2:- Vertex.
$x-$coordinate of the vertex is $x=-\frac{b}{2a}=-\frac{(-7)}{2(2)}=\frac{7}{4}$.
Substitute the value of $x$ into the function.
$\Rightarrow f(\frac{7}{4})=2(\frac{7}{4})^2-7(\frac{7}{4})-4$
Simplify.
$\Rightarrow f(\frac{7}{4})=\frac{49}{8}-\frac{49}{4}-4$
The LCD is $8$.
Multiply the numerator and the denominator to form LCD at the denominator.
$\Rightarrow f(\frac{7}{4})=\frac{49}{8}-\frac{98}{8}-\frac{32}{8}$
Add all numerators.
$\Rightarrow f(\frac{7}{4})=\frac{49-98-32}{8}$
Simplify.
$\Rightarrow f(\frac{7}{4})=-\frac{81}{8}$
The vertex is $\left(\frac{7}{4},-\frac{81}{8}\right)$.
Step 3:- $x-$intercepts.
Replace $f(x)$ with $0$ into the given function.
$\Rightarrow 0=2x^2-7x-4$
Use quadratic formula, we have $a=2,b=-7$ and $c=-4$.
$\Rightarrow x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$
Substitute all values.
$\Rightarrow x=\frac{-(-7)\pm\sqrt{(-7)^2-4(2)(-4)}}{2(2)}$
Simplify.
$\Rightarrow x=\frac{7\pm\sqrt{49+32}}{4}$
$\Rightarrow x=\frac{7\pm\sqrt{81}}{4}$
$\Rightarrow x=\frac{7\pm9}{4}$
Separate the fractions.
$\Rightarrow x=\frac{7+9}{4}$ or $ \frac{7-9}{4}$
Simplify.
$\Rightarrow x=\frac{16}{4}$ or $ x=\frac{-2}{4}$
$\Rightarrow x=4$ or $ x=-\frac{1}{2}$
The $x-$intercepts are $4$ and $-\frac{1}{2}$. The parabola passes through $(4,0)$ and $\left(-\frac{1}{2},0\right)$.
Step 4:- $y-$intercept.
Replace $x$ with $0$ in the given function.
$\Rightarrow f(0)=2(0)^2-7(0)-4$
Simplify.
$\Rightarrow f(0)=-4$
The $y-$intercept is $-4$. The parabola passes through $(0,-4)$.
Step 5:- Graph.
Use the points:- vertex, $x-$intercepts and $y-$intercept to draw a parabola.
The axis of symmetry is $x=-\frac{3}{2}$.
$A=\left(\frac{7}{4},-\frac{81}{8}\right)$
$B=(4,0)$
$C=\left(-\frac{1}{2},0\right)$
$D=(0,-4)$.
From the graph the range of the function is
$\left[-\frac{81}{8},\infty\right)$.
