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Factored Form

Site: Gladwin
Course: Michigan Algebra I Sept. 2012
Book: Factored Form
Printed by: Guest user
Date: Thursday, November 21, 2024, 8:44 AM

Description

Factored Form

Table of contents

Quadratic Equations

When a quadratic function crosses the x-axis, those x-values are called the roots or zeros of the function. For a quadratic function that has real roots,QuadEq1-1 and QuadEq1-2, the factored form is given as: QuadEq1-3; where a is a real number that works the same way as the a-value in the last lesson.

For example: See how the graph of QuadEq1-4 intersects the x-axis at -1 and 5.

QuadEq1-5

The factored form of a quadratic equation also tells you where the x-intercepts are located. The x-intercepts of the equation are the x-values that will make y = 0.


Graphing

To graph a quadratic equation in factored form, first identify the x-intercepts. Use the a-value to determine if the graph will open up or down. The vertex of the parabola can be found exactly midway between the two x-intercepts, using the symmetry of the parabola. It is then possible to find the y-intercept by substituting x = 0 into the equation and solving. Finally, use the symmetry of the graph to find a point on the other side of the axis of symmetry.


Example

Make a graph of the following function: f(x) = (x + 3) (x - 1)

Step 1. Determine the x-intercepts.

0 = (x + 3)(x - 1)

x + 3 = 0 or x - 1 = 0

x = -3 or x = 1

Step 2. Find the a-value.

The a-value is positive one, so the graph opens up.

Step 3. Calculate the vertex of the graph.

The midpoint of the x-intercepts is: QuadEqEx1-1 .

The y-coordinate is QuadEqEx1-2

The vertex is (-1, -4).

Example Continued

Step 4. Find the y-intercept and the symmetry point.

The y-intercept is QuadEqEx1-3

The y-intercept is (0, -3).

The symmetry point is (-2, -3).

Step 5. Plot the points and sketch the curve.


Interactive Activity

To see how quadratic graphs change as factors change, select the following link:

Factored Form Graphing

Practice

Factored Form Graphing Worksheet

*Note: If Google Docs displays “Sorry, we were unable to retrieve the document for viewing,” refresh your browser.

Answer Key

Factored Form Graphing Answer Key

*Note: If Google Docs displays “Sorry, we were unable to retrieve the document for viewing,” refresh your browser.

Writing Equations

Since the factored form shows the x -intercepts of the function, it is possible to write an equation in factored form if you know the x -intercepts. Substitute values into the form: WritingEquation1 and simplify.
Example Model the quadratic function graphed below using an equation in factored form.


WritingEx1-1



Step 1. Find the x-intercepts.

The x-intercepts of the graph are (0, 0) and (4, 0).

Step 2. Substitute the x-intercepts into the general form.

WritingEx1-2

Step 3. Substitute another point from the graph into the general form and solve for the a-value.

WritingEx1-3

Step 4. Substitute the a-value into the formula.

WritingEx1-4

Practice

Writing Factored Form Equations Worksheet

*Note: If Google Docs displays “Sorry, we were unable to retrieve the document for viewing,” refresh your browser.

Answer Key

Writing Factored Form Equations Answer Key

*Note: If Google Docs displays “Sorry, we were unable to retrieve the document for viewing,” refresh your browser.

Sources

Embracing Mathematics, Assessment & Technology in High Schools; A Michigan Mathematics & Science Partnership Grant Project

Hot Math, "Factored Form Applet." http://hotmath.com/util/hm_flash_movie. html?movie=/learning_activities/interactivities/quadratics.swf&title=Quadratics%20in%20Factored%20Form (accessed 08/05/2010).

"What the Factored Form Can Tell You About a Graph." http://algebra-tutoring.com/solving-quadratic-equations-6.htm (accessed 8/05/2010).