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Explorations in Core Math - Algebra 1: Online Textbook Help10 chapters | 121 lessons

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.

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.

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.

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.

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

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

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 (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.

(*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.

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*.

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

To graph a quadratic function you will need to know a few other points. The *y*-intercept is the point where the graph crosses the *y* axis. It is located at the point (0,*c*) using the standard form. The vertex of a quadratic is the highest or lowest point on the graph. The *x* value of the vertex is found using -*b*/2*a* in standard form. Once you have determined the *x* value of the vertex, you plug the *x* value into the function to determine the *y* value. The graph will open up or down depending on the value of *a*. If *a* is negative, the parabola opens down; if it is a positive number, it will open up.

To graph a quadratic equation you will need to determine the zeros, the y-intercept, the vertex, and the direction it opens. You will then plot those points on an *x,y* coordinate grid and fill in the rest of the points with a smooth parabola.

Graph the function: x2-4x=12

- Put into standard form: x2-4x-12=0
- Determine
*a,b,and c*:*a*=1*b*=-4*c*=-12 *Y*-intercept (0,c): (0,-12)- Opens up or down:
*a*is positive so opens up - Vertex -b/2a: 4/2(1) = 2, plug in 2, (2)2-4(2)-12=-16 Vertex (2,-16)
- Find zeros: Two numbers whose product is -12 and sum is -4. They are -6,+2
- Factored Form: (x-6)(x+2)=0
- Zeros at: (6,0) and (-2,0)

To graph a quadratic equation it is very helpful to know where the zeros of the function are. If you put the equation into **factored form**, you can easily solve for *x* using the **zero product rule**. Determine the location of the vertex, and the y-intercept. You now have enough points to draw a **parabola** connecting those points.

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Explorations in Core Math - Algebra 1: Online Textbook Help10 chapters | 121 lessons

- Quadratic Functions: Examples & Formula 3:56
- Using Quadratic Functions to Model a Given Data Set or Situation 5:53
- Quadratics: Equations & Graphs
- Graphing Quadratic Functions by Factoring
- How to Solve a Quadratic Equation by Graphing
- How to Solve a Quadratic Equation by Factoring 7:53
- Zero Product Property: Definition & Examples 4:11
- How to Solve Quadratics with Complex Numbers as the Solution 5:59
- Write the Standard Form of an Equation by Completing the Square 6:21
- How to Use the Quadratic Formula to Solve a Quadratic Equation 9:20
- How to Solve Quadratics That Are Not in Standard Form 6:14
- Discriminant: Definition & Explanation
- Applying Quadratic Functions to Motion Under Gravity & Simple Optimization Problems 7:42
- Solving Nonlinear Systems with a Quadratic & a Linear Equation
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