Instructional Unit 9

Factored Form of Quadratics Functions/Looking at Zeros Graphically

by Sandy Cederbaum

Thus far we have explored the vertex form and the standard (polynomial form) of a quadratic function. There is a third form that is sometimes useful in the study of quadratic functions called factored form. We can see the value of the factored form of a quadratic function in the following example.

Given the function , we can write the f(x) in factor form . If we are interested in finding the zeros (x-intercepts) of the function, we can substitute 0 for f(x) and use the Zero Products Property to solve the equation as follows;

Use your graphing utility to graph the linear functions and in the same viewing window. Next graph as the product of and in the same window. Graph

Write a sentence or two describing the relationship between the linear function, the quadratic function and the zeros of each. Create your own quadratic functions of the form a(x-d)(x-e). Start with a fixed value for a and vary both d and e. Use positive and negative values for d and e. Also try using some non-integer values for d and e. Do the sentences you wrote describing the relationship between the linear functions, quadratic functions and zeros still hold?

Now fix the values of d and e. and vary a. Write a sentence or two to describe the similarities and differences in these graphs. Below is an example where d =2, e = -1 and a takes on integer values from -3 to 3.

Following is a QuickTime Movie that illustrates this same point for all real values of a from -5 to 5. What does this suggest about the number of different parabolic functions that can have the zeros x=-1 and x=2?

Here are several examples of quadratic functions that are factorable. They are organized by level of difficulty (1 being the least difficult). Factor each function and determine the zeros of each.

Factors

In the end, we can factor, within reason, any quadratic function that has real zeros using a graphical approach or the quadratic formula. Explain how you might go about factoring the following quadratic function.

We have now explored three different algebraic representations of quadratic functions shown in the table below

 Vertex Form Standard Form Factored Form

Each of these forms has its advantages and disadvantages in answering the following five questions outlined in Unit 6

1. Find the y-intercept of the function.

2. Find the value of y (or the dependent variable) when x (the independent variable) is specified.

3. Find the x-intercepts of the function.

4. Find the vertex (or find the coordinates of the maximum or minimum value on the graph) of the function.

5. Find the value of x (or the independent variable) when y (the dependent variable) is specified.

As your final task in this unit, consider the five questions above. Write a brief paragraph describing which form of a quadratic function would be preferred in seeking the answer to each question above.

The instructional units that you have explored thus far have all dealt with quadratic functions using an algebraic approach. In unit 10, you will discover that there is also a geometric interpretation for a parabola. This geometric interpretation is helpful in understanding some of the physical properties of parabolas that enable us to use them in every day life.

Move on to Unit 10 - A Geometric Interpretation of a Parabola