Learning Objectives
- Graph quadratic functions in intercept form [latex] f(x)=a(x-p)(x-q) [/latex].
- Use the Zero-Product Property to solve quadratic equations in intercept form.
- Find the [latex] x [/latex]-intercepts, [latex] y [/latex]-intercept, axis of symmetry, and vertex of a quadratic function in intercept form.
Graphing Quadratics in Intercept Form [latex] f(x)=a(x-p)(x-q) [/latex]
In this section we will be graphing quadratic functions that are written in intercept form. When quadratic functions are written in this form, it is easy to identify the [latex] x [/latex]-intercepts. In order to graph the parabola, we will also need to determine the vertex, [latex] y [/latex]-intercept, and the axis of symmetry. For reference, Figure 1 illustrates important features of a parabola that you learned about in the last section.
Figure 1
To find the [latex] x [/latex]-intercepts of a quadratic equation in intercept form, we first need to learn about the Zero-Product Property.
What if we told you that we multiplied two numbers together and got an answer of zero? What could you say about the two numbers? Could they be [latex]2[/latex] and [latex]5[/latex]? Could they be [latex]9[/latex] and [latex]1[/latex]? No! When the result (answer) from multiplying two numbers is zero, that means that one of them had to be zero. This idea is called the Zero-Product Property, and it is useful for solving polynomial equations that can be factored.
The Zero-Product Property
The Zero-Product Property states: If the product of two numbers is zero, then at least one of the numbers is zero. That is
where [latex] a [/latex] and [latex] b [/latex] are real numbers or algebraic expressions.
ExAMPLE
- Solve [latex] (x-4)(x+7)=0 [/latex] using the Zero-Product Property.
- Solve [latex] (x+3)(2x-5)=0 [/latex] using the Zero-Product Property.
ExAMPLE
Graph the parabola [latex] f(x)=(x+2)(x+4) [/latex] by first finding the [latex] x [/latex]-intercept(s), [latex] y [/latex]-intercept, axis of symmetry, and the vertex.
ExAmple
Find the [latex] x [/latex]-intercept(s), [latex] y [/latex]-intercept, axis of symmetry, and the vertex of [latex] f(x)=-x(x-6) [/latex]. Then graph the parabola.
ExAMPLE
Let [latex] f(x)=x(x+8) [/latex]. Fill in the table and graph the function.
| [latex] x [/latex] | [latex] f(x)=x(x+8) [/latex] |
|---|---|
ExAMPLE
Let [latex] f(x)=(x+1)(5-x) [/latex]. Fill in the table and graph the function.
| [latex] x [/latex] | [latex] f(x)=(x+1)(5-x) [/latex] |
|---|---|
Below, think about the function given in regards to graphing it. It is written in general form which will be discussed more in the next section. When we try to graph it, we find something interesting.
THink About it
Consider the function [latex] g(x)=x^2+5x+6 [/latex].
What are some efficient [latex] x [/latex]-values we can choose to find important features of the parabola?
If we let [latex] \color{Plum}{x=0} [/latex], we can find the [latex] \color{Plum}{\textsf{y-intercept}} [/latex].
[latex] \begin{align} g(\color{Plum}{\textbf{0}}\color{black}{)} &= (\color{Plum}{\textbf{0}}\color{black}{)^2+5(}\color{Plum}{\textbf{0}}\color{black}{)+6} \\ &= 6 \end{align} [/latex]
The [latex] y [/latex]-intercept is [latex] (0,6) [/latex]
To find the [latex] x [/latex]-intercept(s), let [latex] y=0 [/latex] and solve for [latex] x [/latex].
[latex] 0=x^2+5x+6 [/latex]
How can we solve this equation for x?
It looks like we might need something else to help us find the [latex] x [/latex]-intercept(s).
How can we write this equation in intercept form?
We discovered earlier in this lesson that it is easy to find the [latex] x [/latex]-intercept(s) in intercept form, using the Zero-Product property. Let’s skip finding the [latex] x [/latex]-intercept(s) and work on finding the vertex.
We learned that the vertex is exactly half-way between the [latex] x [/latex]-intercepts because of the symmetry of parabolas. We were not able to find the [latex] x [/latex]-intercepts so we cannot find the vertex using this method. If we knew the vertex, we could also find the axis of symmetry.
We could choose random values for [latex] x [/latex] and determine the [latex] y [/latex]-values but these [latex] x [/latex]-values we choose will not necessarily help us find the important features of this parabola.
It is apparent that it is more difficult to find the [latex] x [/latex]-intercept(s), vertex, and axis of symmetry if the function is not in intercept form. Intercept form is sometimes called factored form. Learning more about factoring will be useful for us. Factoring is a great skill to develop and is helpful for this course as well as subsequent courses.
Summary
- [latex] x [/latex]-intercepts are easy to find when function is written in intercept form [latex] f(x)=(x-p)(x-q) [/latex] and harder to determine in general form [latex] f(x)=ax^2+bx+c [/latex].
- The Zero-Product Property can be used to solve equations of the form [latex] (x+a)(x+b)=0 [/latex] by setting each factor equal to 0, [latex] x+a=0 [/latex] and [latex] x+b=0 [/latex].
- The vertex can be found in intercept form by finding the point that is half-way between the [latex] x [/latex]-intercepts.
- Th axis of symmetry is the vertical line that divides the parabola into two equal halves that passes through the vertex. It is easy to find when you know the vertex.
- The [latex] y [/latex]-intercept can be found easily by letting [latex] x=0 [/latex] and solving for [latex] y [/latex].
- Parabolas are “U”-shaped graphs with a smooth curve. The graph should be rounded at the vertex. Parabolas are NOT “V”-shaped graphs.
