6.1: Introduction to Quadratic Functions

Prior Lesson | Next Lesson

Objectives:

• Recognize quadratic functions in standard and factored forms
• Find $x$-intercepts of quadratic functions by using factors

Important Items

Definitions:

standard form, parabola, quadratic

Notes: Students have seen quadratic functions, factoring, and the quadratic formula before, but most likely have not mastered these topics. This means that you should rely on the students to generate the basic ideas but add your insights and understanding to the students previous knowledge. It is a mistake to assume that nobody has seen this material before and pretend this is a first presentation. We are trying to help the students understand their previous knowledge. Throughout this chapter, really try to emphasize the different forms of a quadratic function and why some forms are better for answering various questions than the other forms.

---

Lesson Guide

Warm-Up

Have students do Problems 1 and 2.

Introduce quadratic functions and standard form

Introduce the shape of quadratic functions (parabolas) using several real world examples. These can vary widely and can include things like water fountains, the graph of the height of a ball that is thrown straight up into the air, $y=x^2$ graphed on axes, some parabolas concave up/down, and some with 0, 1, 2 $x$-intercepts. Say this shape is called a {\em parabola}, and that functions whose graphs have this shape are called {\em quadratic functions}.

• Example:

Parabolas show up in real life often because of the effects of gravity (like we see in Angry Birds and the water fountain). If a ball is thrown upward from the top of a building, even if it just goes straight up and down, then its vertical height after $t$ seconds is given by a quadratic function, for example, $h(t) = -16t^2 + 32t + 128$. By the end of class today, we'll be able to find the height from which the ball was thrown and the time when the ball reaches the ground. After tomorrow's class, we'll be able to find the maximum height the ball reaches.

• Example:

The standard form for a quadratic function is $y = ax^2 + bx + c$, where $a, b$, and $c$ are constants and $a \neq 0$. 
The graph of a quadratic function is called a parabola.
 

Ask students to recall from Chapter 1 how we find the $y$-intercept of a function. (In general, to find the $y$-intercept of a function $f(x)$, we evaluate $f(0)$.)

Have students recall Problem 1 on the worksheet. Ask students to see if they note a relationship between the standard form of a quadratic function and the $y$-value of the $y$-intercept.

Point out that we can read the $y$-intercept right off the standard form. Ask, ``What does \textbf{c} in standard form tell us about the graph of the quadratic function $f(x) = ax^2 + bx + c$? It's the $y$-intercept.

Also note that the function has two $x$-intercepts. Remind students to be careful when taking square roots!

If $y = f(x) = ax^2 +bx +c$ is a quadratic function in standard form, then the graph of $f$ has a $y$-intercept at $(0,c)$. Also, if $a > 0$, 
then the parabola opens upward, and if $a<0$, the parabola opens downward.

Have students do Problem 3. Explain to students that just because they see something like $h(x)=-(x-4)(x+4)$ they should not automatically start distributing but instead to do what the Problem tells them to do. In this case they are looking for standard form and so it makes sense to distribute, but in other cases it might make sense to leave the equation alone. This is a really difficult concept for students to understand. Don't worry if at first they don't follow, just keep telling the students that mathematical forms have different uses in different contexts and the context is what they need to work on understanding.

Find $x$-intercepts by factoring and quadratic formula

Although factoring is done in 100A and is something all students will have seen in high school, students usually need a review of factoring quadratics. Once in factored form, students have the tendency to want to set the quadratic expression equal to zero and then solve the equation. Emphasize to students that this is what we must do to find the $x$-intercepts of a quadratic function, but simply factoring an expression is different than solving an equation.

Note: Students will come in with a probably-correct-but-mysterious-and-misunderstood method to find the zeros of quadratics, but we want them to learn the $ac$-method (shown below). This standardizes our expectations and gets rid of some of the nonsensical methods like "slide and divide." You can adjust the following example to fit your style, but make sure that the "algorithm" you teach is essentially the same.

• Example:

Find the $x$-intercepts of $f(x) = x^2 - 3x-4$.

$a = 1$, $c = -4$, so $ac = -4$. We need factors of $-4$ that add to give $-3$, so we choose $1$ and $-4$. Then, \[\begin{array}{rl} x^2 - 3x - 4 & = x^2-4x + x - 4 & = x(x-4) + 1(x-4) & = (x+1)(x-4). \end{array}\] Now, take $f(x) = (x+1)(x-4) = 0$, and we see that either $x+1 = 0$ or $x-4 = 0$, so $x=-1$ or $x=4$. Thus, $f$ has zeros at $x=-1$ and $x=-4$; equivalently, $f$ has $x$-intercepts $(-1,0)$ and $(4,0)$.

Note that we were able to write $f(x)$ in the form $f(x) = (x+1)(x-4)$. In general, this is called factored form.

The factored form, when it exists, of a quadratic function is 
\[
 f(x) = a(x-r)(x-s),
\]
where $a \neq 0$ is a constant and $r$ and $s$ are the zeros (or $(r,0)$ and $(s,0)$ are the $x$-intercepts) of $f$.

So for our previous example, we have $a = 1$, $r = -1$ and $s=4$. Note that $-1$ and $4$ are the two zeros of $f$.

Have students do Problem 4. Note that part (a) is already in factored form.

Introduce the quadratic formula

Have students solve Problem 5(a). Ask them what went wrong when trying to factor the function. Ask if anyone knows of another way to find the zeros of a quadratic function; hopefully someone will remember the quadratic formula.

In general, the zeros of a quadratic function $f(x) = ax^2+bx+c$, if they exist, are given by
\[
x = \dfrac{-b \pm \sqrt{b^2-4ac }}{2a}.
\]

Feel free to do an example if your class seems nauseous at the sight of the formula.

Have students do Problems 5-6.

Have students try Problem 7. Try not to give too big of hints on this. The students now have factored form and standard form. See if they can decide how to put all of these together to find the formula.