Topics
7. Quadratic Functions
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United States of AmericaFL
Grade 912

7.03 Quadratic functions in factored form

Lesson
Practice
Lesson

Concept summary

One way to represent quadratic functions is using the factored form. This form allows us to identify the x-intercepts, direction of opening, and scale factor of the quadratic function.

\displaystyle y=a(x-x_1)(x-x_2)
\bm{x_1, \,x_2}
x-values of the x-intercepts
\bm{a}
The scale factor, so tells us about the shape of the graph
-4
-3
-2
-1
1
2
3
4
x
-4
-3
-2
-1
1
2
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4
y
If a>0, then the quadratic function opens upwards and has a minimum value.
-4
-3
-2
-1
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x
-4
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-1
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y
If a<0, then the quadratic function opens downwards and has a maximum value.

To draw the graph of a quadratic function, we generally want to find three different points on the graph, such as the x- and y-intercepts. As the graph has a line of symmetry passing through the vertex, we know the vertex lies half way between the two x-intercepts. We can also determine the direction in which the graph opens by identifying if the scale factor, a, is positive or negative.

Worked examples

Example 1

A quadratic function in factored form has the equation:

y=2(x-4)(x+6)

a

State the coordinates of the x-intercepts.

Approach

In the factored form y=a(x-x_1)(x-x_2) the values of x_1 and x_2 are the x-values of the x-intercepts. The y-value of the x-intercepts is y=0.

Solution

The x-intercepts are \left( 4,0 \right) and \left( -6,0 \right).

Reflection

Notice that x+6 is the same as x-(-6).

b

Determine the coordinates of the y-intercept.

Approach

The y-value of the y-intercept is the result when x=0. We can substitute x=0 into the factored form to find this value.

Solution

To find the y-value of the y-intercept:

\displaystyle y\displaystyle =\displaystyle 2(x-4)(x+6)Given quadratic function
\displaystyle y\displaystyle =\displaystyle 2(0-4)(0+6)Substitute in x=0
\displaystyle y\displaystyle =\displaystyle 2(-4)(6)Simplify the factors
\displaystyle y\displaystyle =\displaystyle -48Simplify

The y-intercept is \left( 0,-48 \right).

c

Determine the coordinates of the vertex.

Approach

The vertex lies on the axis of symmetry, so the x-coordinate of the vertex will be the average of 4 and -6. We can then substitute this value into the funtion to find the y-coordinate.

Solution

To find the x-coordinate:

The average of 4 and -6 is half way between them. We can calculate that \dfrac{4+(-6)}{2}=-1, so the x-coordinate of the vertex and the axis of symmetry is x=-1.

To find the y-coordinate:

\displaystyle y\displaystyle =\displaystyle 2(x-4)(x+6)Given quadratic function
\displaystyle y\displaystyle =\displaystyle 2(-1-4)(-1+6)Substitute in x=-1
\displaystyle y\displaystyle =\displaystyle 2(-5)(5)Simplify the factors
\displaystyle y\displaystyle =\displaystyle -50Simplify

The vertex is \left( -1,-50 \right).

d

Draw the graph of the function.

Approach

The scale factor is 2 which is positive, so the graph will open up. We can draw the graph through any three points that we know are on it.

Solution

-8
-6
-4
-2
2
4
6
8
x
-60
-50
-40
-30
-20
-10
10
20
y

Reflection

Any three points is enough to draw the graph, but knowing where the vertex is can make it easier since the vertex is on the axis of symmetry.

Example 2

Consider the graph of a quadratic function:

-4
-3
-2
-1
1
2
3
4
x
-4
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-2
-1
1
2
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y
a

Identify the coordinates of the x- and y-intercepts of the function.

Solution

The x-intercepts are \left( -2,0 \right) and \left( 3,0 \right).

The y-intercept is \left( 2,0 \right).

b

Find the equation of the quadratic function in factored form.

Solution

Since the x-values of the x-intercepts are -2 and 3, we know that the factored form will be:

y=a(x+2)(x-3)

for some value of a. We can find a by substituting in the coordinates of the y-intercept into the function.

To find a:

\displaystyle y\displaystyle =\displaystyle a(x+2)(x-3)Factored form
\displaystyle 2\displaystyle =\displaystyle a(0+2)(0-3)Substitute in the y-intercept
\displaystyle 2\displaystyle =\displaystyle a(2)(-3)Simplify the factors
\displaystyle 2\displaystyle =\displaystyle -6aSimplify
\displaystyle -\frac{1}{3}\displaystyle =\displaystyle aDivide both sides by -6

The equation of the quadratic function in factored form:

y=-\frac{1}{3}(x+2)(x-3)

Outcomes

MA.912.AR.1.3

Add, subtract and multiply polynomial expressions with rational number coefficients.

MA.912.AR.1.7

Rewrite a polynomial expression as a product of polynomials over the real number system.

MA.912.AR.3.5

Given the x-intercepts and another point on the graph of a quadratic function, write the equation for the function.

MA.912.AR.3.7

Given a table, equation or written description of a quadratic function, graph that function, and determine and interpret its key features.

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