Task Sets: Graph polynomial functions - Common Core Math - HS Functions

A

Recall / Warm-Up

1.

What are the zeros of $f(x) = (x - 3)(x + 5)$ ?

A.

$x = 3$ and $x = -5$

B.

$x = -3$ and $x = 5$

C.

$x = 3$ and $x = 5$

D.

$x = -3$ and $x = -5$

2.

Factor $f(x) = x^3 - 9x$ . What are the zeros?

A.

$x = 0$ only

B.

$x = 3$ and $x = -3$

C.

$x = 0$ , $x = 3$ , and $x = -3$

D.

$x = 9$ and $x = -9$

3.

Which of the following is TRUE about polynomial functions?

A.

Polynomial graphs can have vertical asymptotes where the function is undefined.

B.

Polynomial graphs are smooth, continuous curves defined for all real numbers.

C.

A polynomial of degree $n$ has exactly $n$ turning points.

D.

The end behavior of a polynomial depends on its middle terms.

B

Fluency Practice

C

Varied Practice

D

Word Problems

1.

A polynomial is given in factored form: $f(x) = -(x + 3)(x - 1)^2(x - 5)$ .

(a) List all zeros and state the multiplicity of each.
(b) At each zero, does the graph cross or bounce?
(c) What is the degree? What are the maximum number of turning points?
(d) Describe the end behavior.

2.

A polynomial function $g$ has degree 4, a negative leading coefficient, zeros at $x = -2$ (multiplicity 2), $x = 1$ (multiplicity 1), and $x = 3$ (multiplicity 1).

1.

Write a factored-form equation for $g$ that matches the given information.

2.

Describe the graph behavior at each zero and the end behavior. How many turning points does the graph have at most?

3.

A polynomial function is given: $f(x) = x^4 - 5x^2 + 4$ .

(a) Factor completely and identify all real zeros.
(b) State the end behavior and the maximum number of turning points.
(c) Describe the crossing/bouncing behavior at each zero.

E

Error Analysis

1.

A student is asked: "How many turning points does $f(x) = x^5 - 3x^3 + x$ have?" The student answers: "It has degree 5, so it has exactly 5 turning points."

What is wrong with the student's reasoning?

A.

The maximum number of turning points is $n - 1$ , not $n$ . A degree-5 polynomial has at most 4 turning points — it may have fewer.

B.

The degree of $f$ is not 5. Check the highest-degree term.

C.

The student is correct — a degree-5 polynomial always has exactly 5 turning points.

D.

A polynomial can have an unlimited number of turning points.

2.

A student graphs $f(x) = (x - 3)^2(x + 1)$ and draws the graph crossing the $x$ -axis at both $x = 3$ and $x = -1$ .

What mistake did the student make at $x = 3$ ?

A.

At $x = 3$ , the factor $(x-3)^2$ has multiplicity 2 (even), so the graph should bounce — not cross.

B.

The student drew the wrong $x$ -intercepts. The zeros should be $x = -3$ and $x = 1$ .

C.

The crossing at $x = -1$ is wrong; the graph should bounce there instead.

D.

There is no error — all zeros produce crossing behavior on graphs.

F

Challenge / Extension

1.

Find a polynomial of degree 5 with exactly two distinct $x$ -intercepts, where the graph bounces at one and crosses at the other. Write the function in factored form and describe all key features (zeros, multiplicities, end behavior, maximum number of turning points).

2.