Lesson 6 Puzzling Over Polynomials Practice Understanding

Learning Focus

Combine pieces of information about polynomials to write equations and graph them.

Identify features of polynomials from equations and graphs.

How do polynomial roots and end behavior work together to make equations and graphs?

Open Up the Math: Launch, Explore, Discuss

Each of these polynomial puzzles given contains a few pieces of information. Your job is to use that information to complete the puzzle. Occasionally, you may find a missing piece that you can fill in yourself. For instance, although some of the roots are given, you may decide that there are others that you can fill in. When you need to graph a function, imagine what it will look like before using technology. Then use technology to graph the function and see how close your idea was to the actual function.

1.

Function:

$f (x) = 2 (x - 1) {(x + 3)}^{2}$

End behavior:

As $x \to - \infty$ , $f (x) \to$ .

As $x \to \infty$ , $f (x) \to$ .

Roots (with multiplicity):

Value of leading coefficient:

Domain:

Range: All real numbers.

Graph:

Pause and Reflect:

2.

Function (in factored form):

Function (in standard form) :

End behavior:

As $x \to - \infty$ , $f (x) \to$ .

As $x \to \infty$ , $f (x) \to$ .

Roots (with multiplicity):

$- 2$ , $1$ (multiplicity $2$ )

Value of leading coefficient: $- 2$

Degree: $3$

Graph:

3.

Function (in factored form):

Function (in standard form):

End behavior:

As $x \to - \infty$ , $f (x) \to \infty$ .

As $x \to \infty$ , $f (x) \to$ .

Roots (with multiplicity):

$3$ (mult. $1$ ), $- 1$ (mult. $2$ ), $0$ (mult. $2$ )

Value of leading coefficient: $- 1$

Domain:

Range:

Graph:

4.

Function:

End behavior:

As $x \to - \infty$ , $f (x) \to$ .

As $x \to \infty$ , $f (x) \to$ .

Roots (with multiplicity):

Value of leading coefficient: $1$

Domain:

Range:

Other: $f (0) = 16$

Graph:

5.

Function (in factored form):

End behavior:

As $x \to - \infty$ , $f (x) \to$ .

As $x \to \infty$ , $f (x) \to$ .

Roots (with multiplicity):

$2 + i$ , $4$ , $0$ ,

Value of leading coefficient: $1$

Degree: $4$

Graph:

6.

Function (in standard form):

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

Function (in factored form):

End behavior:

As $x \to - \infty$ , $f (x) \to$ .

As $x \to \infty$ , $f (x) \to$ .

Roots (with multiplicity):

$- 2$ ,

Domain:

Range:

Graph:

7.

Function (in standard form):

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

Function (in factored form):

End behavior:

As $x \to - \infty$ , $f (x) \to$ .

As $x \to \infty$ , $f (x) \to$ .

Roots (with multiplicity):

Domain:

Range:

Graph:

Ready for More?

Write your own puzzle to trade with a partner. Try coming up with one or all of these:

A puzzle for a degree $3$ polynomial with $1$ real root. The solver needs to find the equation and the graph.
A puzzle for a degree $4$ polynomial with $2$ complex roots. The solver needs to find the equation and the graph.
A puzzle for a degree $3$ polynomial where the solver is given a graph and needs to find the roots and write the equation in factored form.

Takeaways

Tips for finding equations and graphs of polynomials:

Lesson Summary

In this lesson, we put together everything we have learned in the unit to write equations and graph polynomials. We wrote equations given roots and found roots given equations using the relationship between roots and factors. We graphed polynomials using the roots and end behavior to predict the shape of the curve.

Retrieval

1.

Divide. Leave answer in lowest terms. (Assume no denominator equals $0$ .) $\frac{x \cdot x \cdot x \cdot y (6 x - 5) (x + 8)}{- 5 \cdot x \cdot x \cdot y \cdot y (x + 8) (6 x + 5)}$

2.

Write the polynomial function in standard form given the leading coefficient and the roots of the function.

Leading coefficient: $- 3$
Roots: $1 + 5 i$ , $1 - 5 i$