We are asked to solve the given quadratic equation by graphing. There are three steps to solving a quadratic equation by graphing.

Write the equation in standard form, $a x^{2} + b x + c = 0 .$
Graph the related function $y = a x^{2} + b x + c .$
Find the $x -$ intercepts, if any.

The solutions of

a x^{2} + b x + c = 0

are the

x -

intercepts of the graph of

y = a x^{2} + b x + c .

Our equation is already written in standard form. Let's identify the function related to the equation.

Equation : Related Function : x^{2} - x - 2 = 0 y = x^{2} - x - 2

Graphing the Related Function

To draw the graph of the related function written in standard form, we must start by identifying the values of

a,

b,

and

c .

y = x^{2} - x - 2 \Leftrightarrow y = 1 x^{2} + (- 1) x + (- 2)

We can see that

a = 1,

b = - 1,

and

c = - 2 .

Now, we will follow four steps to graph the function.

Find the axis of symmetry.
Calculate the vertex.
Identify the $y -$ intercept and its reflection across the axis of symmetry.
Connect the points with a parabola.

Finding the Axis of Symmetry

The axis of symmetry is a vertical line with the equation

x = - \frac{b}{2 a} .

Since we already know the values of

a

and

b,

we can substitute them into the formula.

x = - \frac{b}{2 a}

SubstituteII

$a = 1$ , $b = - 1$

x = - \frac{- 1}{2 ( 1 )}

▼

Simplify right-hand side

MultByOne

$a \cdot 1 = a$

x = - \frac{- 1}{2}

RemoveNegFracAndNum

$- \frac{- a}{b} = \frac{a}{b}$

x = \frac{1}{2}

CalcQuot

Calculate quotient

x = 0.5

The axis of symmetry for the parabola is a vertical line with the equation

x = 0.5 .

Calculating the Vertex

To calculate the vertex, we need to think of

y

as a function of

x,

y = f (x) .

We can write the expression for the vertex by stating the

x -

and

y -

coordinates in terms of

a

and

b .

Vertex : (- \frac{b}{2 a}, f (- \frac{b}{2 a}))

Note that the formula for the

x -

coordinate is the same as the formula for the axis of symmetry, which is

x = 0.5 .

Therefore, the

x -

coordinate of the vertex is also

0.5 .

To find the

y -

coordinate, we need to substitute

0.5

for

x

in the given equation.

y = x^{2} - x - 2

Substitute

$x = 0.5$

y = 0.5^{2} - 0.5 - 2

▼

Simplify right-hand side

CalcPow

Calculate power

y = 0.25 - 0.5 - 2

SubTerms

Subtract terms

y = - 2.25

We found the

y -

coordinate and now we know that the vertex is

(0.5, - 2.25) .

Identifying the $y -$ intercept and its Reflection

The $y -$ intercept of the graph for a quadratic function written in standard form is given by the value of $c .$ The point where our graph intercepts the $y -$ axis is $(0, - 2) .$ Let's plot this point and its reflection across the axis of symmetry.

Connecting the Points

We can now draw the graph of the function. Since $a = 1,$ which is positive, the parabola will open upwards. Let's connect the three points with a smooth curve.

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