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Graphing Quadratic Functions by Factoring

Instructor: David Karsner
To graph a quadratic function, it is very helpful to know the zeros of that function. One means of finding the zeros is to factor the function and then use the linear factors to solve for the zeros.

Golf Ball on Tee
GolfOnTee

If you were to swing at a golf ball and then record its height until it landed once again; you would be creating the graph of a quadratic function. The zeros of this function would be when the ball is touching the ground. It would be touching the ground at the start, before you hit it, and again when it has landed, hopefully a hole in one. It is possible to graph these functions using several different means. This lesson will describe how to graph a quadratic function by using the zeros (found through factoring) and several other key points.

Graphing a Quadratic Function

A quadratic function is a polynomial that has a degree of two (the largest exponent is a two). The graphs of these functions will always look like a u or an upside down u (the graph is called a parabola). To graph quadratic functions accurately you will need to know several things about them. You should know the zeros (also known as the x-intercepts), the y-intercepts, the vertex, and whether the graph opens up or opens down.

Graphs of Quadratic Functions
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Putting it in Standard Form

The standard form of a quadratic equation will look like ax2+bx+c=0. The values of the a,b,and c will tell us several important pieces of information. In the equation -3x2+5x-8, the value of a is -3, b is 5, and c=-8. This equation is already in standard form, however many times the equation will not be in standard form. If that is the case, you will need to manipulate the equation until it is in standard form.

Example 1

4x2=-3x-7 (add 3x, and 7 to both sides)

4x2+3x+7=0 Now in standard form a=4 b=3 c=7

Example 2

14x2=10 (substract 10 from both sides)

14x2-10=0 (notice there is no x term)

14x2+0x-10=0 Now in standard form a=14 b=0 c=-10

Finding the Zeros Using Factoring

Finding the zeros (the x-intercepts) of a function require the most work. There are several ways of finding these zeros, including the square root property, completing the square, the quadratic formula, and factoring. This lesson will focus on finding the zeros through factoring. Finding the zeros through factoring relies on the zero product rule. The zero product rule states that if two numbers are multiplied together and the product is zero; then one of those numbers has to be zero.

Example

(x-3)(x+4)= 0 Two things multiplied together to give zero.

Either (x-3)=0 or (x+4)=0

Solve both linear equations

x-3=0 (add three to both sides) x=3

x+4=0 (subtract four from both sides) x=-4

Zeros at 3 and -4

On most occasions the quadratic equation will not already be in factored form like the example. Factored form is when the function has been written as the product of two factors equaling zero. You will need to move the equation from standard form to factored form (many quadratic equations will not factor; in this case you will need to use one of the other means of finding the zeros). To factor a quadratic equation that has a 1 for the value of a in standard form, you should find two numbers whose product is c and whose sum is b. a,b,and c are all taken from the standard form.

Example

Find the zeros of x2-5x-14=0

Already in standard form. a=1, b=-5, and c=-14

Looking for two numbers that multiply to -14 and add to -5

Those numbers are -7, and +2

In factored form (x-7)(x+2)=0

Zeros at 7, and -2

Many times the value of a will not be 1. In that case you will need to find two numbers that multiply to a times c, and whose sum is b. You then will take those two numbers and divide them by a.

Example

Find the zeros of 2x2+11x+5=0

Multiply a and c , 2 times 5=10

Two numbers that have a product of 10 and a sum of 11

Those numbers are 1 and 10

Divide each of those numbers by a, a=2

1/2 and 10/2 which is 5

In factored form: (x+1/2)(x+5)=0

The zeros are -1/2 and -5

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