Let's begin by finding the $y -$ intercept, the equation of the axis of symmetry, and the $x -$ coordinate of the vertex. Then, we will graph the function by making a table of values.

Finding the Key Features of The Graph

Consider the general expression of a quadratic function, $f (x) = a x^{2} + b x + c,$ where $a \neq = 0 .$ Let's note three things we can learn from this equation.

The $y -$ intercept is $c .$
The equation of the axis of symmetry is $x = - \frac{b}{2 a} .$
The $x -$ coordinate of the vertex is $- \frac{b}{2 a} .$

We will start by identifying the values of

a,

b,

and

c .

f (x) = 2 x^{2} + 8 x - 3 ⇕ f (x) = 2 x^{2} + 8 x + (- 3)

We can see that

a

=

2,

b

=

8,

and

c

=

- 3 .

Since the

y -

intercept is given by the value of

c,

we know that the

y -

intercept is

- 3 .

Let's now substitute

a = 2

and

b = 8

into

- \frac{b}{2 a}

to find the axis of symmetry and the

x -

coordinate of the vertex.

- \frac{b}{2 a}

SubstituteII

$a = 2$ , $b = 8$

- \frac{8}{2 ( 2 )}

▼

Simplify

Multiply

- \frac{8}{4}

CalcQuot

Calculate quotient

- 2

The equation of the axis of symmetry is

x = - 2,

and the

x -

coordinate of the vertex is

- 2 .

Graphing the Function

Now, let's make a table of values using five points. We want the center point to be the vertex and the remaining points to be symmetric on either side of it. We know that the points will be symmetric if the $x -$ coordinates are equidistant from the axis of symmetry.

$x$	$2 x^{2} + 8 x - 3$	$f (x) = 2 x^{2} + 8 x - 3$
$- 5$	$2 (- 5)^{2} + 8 (- 5) - 3$	$7$
$- 4$	$2 (- 4)^{2} + 8 (- 4) - 3$	$- 3$
$- 2$	$2 (- 2)^{2} + 8 (- 2) - 3$	$- 11$
$0$	$2 (0)^{2} + 8 (0) - 3$	$- 3$
$1$	$2 (1)^{2} + 8 (1) - 3$	$7$

Finally, we will graph the function by plotting the points from the table. Because the graph of a quadratic function is a parabola, we will connect them with a smooth curve. Recall that the axis of symmetry is the line $x = - 2 .$

Exercise