Exercise 17 Page 494

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 .

First, we have to write our equation in the standard form.

x^{2} = 6 x - 9 \Leftrightarrow x^{2} - 6 x + 9 = 0

Now that we have our equation written in the standard form, let's identify the function related to the equation.

Equation : Related Function : x^{2} - 6 x + 9 = 0 y = x^{2} - 6 x + 9

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} - 6 x + 9 \Leftrightarrow y = 1 x^{2} + (- 6) x + 9

We can see that

a = 1,

b = - 6,

and

c = 9 .

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 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

$a = 1$ , $b = - 6$

x = - \frac{- 6}{2 ( 1 )}

▼

Simplify right-hand side

MultByOne

$a \cdot 1 = a$

x = - \frac{- 6}{2}

RemoveNegFracAndNum

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

x = \frac{6}{2}

CalcQuot

Calculate quotient

x = 3

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

x = 3 .

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 = 3 .

Therefore, the

x -

coordinate of the vertex is also

3 .

To find the

y -

coordinate, we need to substitute

3

for

x

in the given equation.

y = x^{2} - 6 x + 9

Substitute

$x = 3$

y = 3^{2} - 6 (3) + 9

▼

Simplify right-hand side

CalcPow

Calculate power

y = 9 - 6 (3) + 9

Multiply

y = 9 - 18 + 9

AddSubTerms

Add and subtract terms

y = 0

We found the

y -

coordinate, and now we know that the vertex is

(3, 0) .

Identifying the $y -$ intercept and its Reflection

The $y -$ intercept of the graph of a quadratic function written in standard form is given by the value of $c .$ Therefore, the point where our graph intercepts the $y -$ axis is $(0, 9) .$ 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.