Lesson 6 To Be Determined Develop Understanding

Jump Start

Which One Doesn’t Belong?

Examine each of the quadratic functions given below and determine which one is not like the others.

$f (x) = (x - 3) (x + 2)$

$f (x) = x^{2} + 3 x - 7$

Reason:

Learning Focus

Write quadratic functions in vertex, factored, and standard form.

Find roots of a quadratic function.

Use the roots of a quadratic function to write the function in factored form.

Can every quadratic function be written in all three of the forms we have studied: standard, vertex, and factored form?

How do solutions of a quadratic relate to the factors of a quadratic function?

Do all quadratic functions have two roots?

Open Up the Math: Launch, Explore, Discuss

Israel and Miriam are working together on a homework assignment. They need to write the equations of quadratic functions from the information given in a table or a graph. At first, this work seemed really easy. However, as they continued to work on the assignment, the algebra got more challenging and raised some interesting questions that they can’t wait to ask their teacher. Work through the following problems from Israel and Miriam’s homework. Use the information in the table or the graph to write the equation of the quadratic function in all three forms. You may start with any form you choose, but you need to find all three equivalent forms.

1.

$x$	$y$
$- 5$	$8$
$- 4$	$3$
$- 3$	$0$
$- 2$	$- 1$
$- 1$	$0$
$0$	$3$
$1$	$8$
$2$	$15$
$3$	$24$
$4$	$35$

Standard form:

Factored form:

Vertex form:

2.

$x$	$y$
$- 5$	$7$
$- 4$	$2$
$- 3$	$- 1$
$- 2$	$- 2$
$- 1$	$- 1$
$0$	$2$
$1$	$7$
$2$	$14$
$3$	$23$
$4$	$34$

Standard form:

Factored form:

Vertex form:

3.

$x$	$y$
$- 5$	$9$
$- 4$	$4$
$- 3$	$1$
$- 2$	$0$
$- 1$	$1$
$0$	$4$
$1$	$9$
$2$	$16$
$3$	$25$
$4$	$36$

Standard form:

Factored form:

Vertex form:

4.

$x$	$y$
$- 5$	$10$
$- 4$	$5$
$- 3$	$2$
$- 2$	$1$
$- 1$	$2$
$0$	$5$
$1$	$10$
$2$	$17$
$3$	$26$
$4$	$37$

Standard form:

Factored form:

Vertex form:

5.

Israel was concerned that his factored form for the function in problem 4 didn’t look quite right. Miriam suggested that he test it out by substituting in some values for $x$ to see if he gets the same points as those in the table. Test your factored form. Do you get the same values as those in the table?

6.

Why might Israel be concerned about writing the factored form of the function in problem 4?

Here are some more examples from Israel and Miriam’s homework.

7.

$x$	$y$
$- 5$	$- 7$
$- 4$	$- 2$
$- 3$	$1$
$- 2$	$2$
$- 1$	$1$
$0$	$- 2$
$1$	$- 7$
$2$	$- 14$
$3$	$- 23$
$4$	$- 34$

Standard form:

Factored form:

Vertex form:

8.

$x$	$y$
$- 5$	$11$
$- 4$	$6$
$- 3$	$3$
$- 2$	$2$
$- 1$	$3$
$0$	$6$
$1$	$11$
$2$	$18$
$3$	$27$
$4$	$38$

Standard form:

Factored form:

Vertex form:

9.

Miriam notices that the graphs of the functions in problems 7 and 8 have the same vertex point. Israel notices that the graphs of the functions in problems 2 and 7 are mirror images across the $x$ -axis. What do you notice about the roots of the functions in problems 7, 8, and 2?

The Fundamental Theorem of Algebra

A polynomial function is a function of the form:

$y = a_{0} x^{n} + a_{1} x^{n - 1} + a_{2} x^{n - 2} + . . . a_{n - 3} x^{3} + a_{n - 2} x^{2} + a_{n - 1} x + a_{n}$

where all of the exponents are positive integers and all of the coefficients $a_{0}$ . . . $a_{n}$ are constants.

As the theory of finding roots of polynomial functions evolved, a 17th century mathematician, Girard (1595-1632) made the following claim which has come to be known as the Fundamental Theorem of Algebra: An $n^{th}$ degree polynomial function has $n$ roots.

10.

In the next unit you will study polynomial functions that contain higher-ordered terms such as $x^{3}$ or $x^{5}$ . Based on your work in this task, do you believe this theorem holds for quadratic functions? That is, do all functions of the form $y = a x^{2} + b x + c$ always have two roots? Examine the graphs of each of the quadratic functions you have written equations for in this task. Do they all have two roots? Why or why not?

Ready for More?

Square roots can often be written in equivalent forms by looking for perfect square factors that can be removed from the radicand. For example, $\sqrt{8} = \sqrt{4 \cdot 2} = 2 \sqrt{2}$ . We can verify that the number $2 \sqrt{2}$ is equivalent to $\sqrt{8}$ by multiplying it by itself and using the commutative property of multiplication to rearrange the factors: $2 \sqrt{2} \cdot 2 \sqrt{2} = 2 \cdot 2 \cdot \sqrt{2} \cdot \sqrt{2} = 2 \cdot 2 \cdot 2 = 8$ . If we allow, at least temporarily, $\sqrt{- 1}$ to be a legitimate factor, show how you might represent $\sqrt{- 8}$ in an equivalent form and then verify that your new representation for $\sqrt{- 8}$ works by multiplying it by itself. Based on this work, rewrite the following radical expressions:

a.

$\sqrt{- 75}$

b.

$\sqrt{- 36}$

c.

$\sqrt{- 32}$

d.

$\sqrt{- 48}$

Takeaways

Observations about the $x$ -intercepts and roots of a quadratic function:

Vocabulary

Lesson Summary

In this lesson, we examined solutions to quadratic equations and connected them with the graph of the function. Solutions for a quadratic equation can be used to write the function in factored form in a process that is the reverse of solving an equation by factoring. We found that when the graph of the quadratic function did not cross the $x$ -axis, the quadratic formula gave solutions that included the square root of a negative number. Although there is not a real number that can be squared to get a negative number, these expressions seemed to work like square roots. Since the Fundamental Theorem of Algebra predicts that all quadratic functions will have two roots, the nature of solutions that involve the square root of a negative number still needs to be resolved.

Retrieval

1.

Write an equivalent form of the square root: $\sqrt{180}$ . (Leave in radical form.)

2.

If the given quadratic function can be factored, factor and provide the $x$ -intercepts. If you cannot factor the equation, use the quadratic formula to find the $x$ -intercepts.

a.

$2 x^{2} - 9 x - 5 = 0$

b.

$2 x^{2} + 6 x - 5 = 0$