Precalculus Polynomial Functions and Their Graphs Worksheet (PDF)

Precalculus math worksheet: Polynomial Functions and Their Graphs – Maths PDF

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This polynomial graphs worksheet for Precalculus covers polynomial functions and their graphs (Common Core F-IF.C.7c, A-APR.B.3, F-IF.B.4), with a free PDF to print.

These 20 problems practice polynomial functions and their graphs (Common Core F-IF.C.7c, A-APR.B.3 and F-IF.B.4). You will find degree, end behavior, zeros, multiplicity and write equations from graphs.

Precalculus · Common Core State Standards: F-IF.C.7c, A-APR.B.3, F-IF.B.4

Polynomial graphs worksheet: exercises 1 to 10

Exercise 1 – Degree and leading coefficient

State the degree and leading coefficient of \(f(x)=5x^{3}-2x^{2}+x-9\).

Exercise 2 – Expand a product

Write \((x-2)(x+3)(x+1)\) in standard form.

Exercise 3 – End behavior, odd degree

Describe the end behavior of \(f(x)=-2x^{5}+x\).

Exercise 4 – End behavior, even degree

Describe the end behavior of \(f(x)=3x^{4}-x^{2}\).

Exercise 5 – Zeros from factors

Find the zeros of \(f(x)=x(x-4)(x+5)\).

Exercise 6 – Multiplicity

For \(f(x)=(x-1)^{2}(x+3)^{3}\), find each zero, its multiplicity and the behavior at the axis. State the degree.

Exercise 7 – Factor out x

Factor \(x^{3}-9x\) completely and find its zeros.

Exercise 8 – Factor by grouping

Factor \(x^{3}+2x^{2}-9x-18\) and find its zeros.

Exercise 9 – Build a polynomial

Write a cubic polynomial with zeros \(-1\), \(2\) and \(4\) and \(f(0)=16\).

Exercise 10 – Read a graph

A cubic graph.

The graph crosses at \(x=-3\) and touches at \(x=1\). Write a factored equation with the least degree, given that the graph rises to the right.

Exercises 11 and up

Exercise 11 – Turning points

(a) What is the greatest number of turning points of a degree-6 polynomial? (b) A graph has 3 turning points. What is the least possible degree?

Exercise 12 – Sketch a polynomial

Axes for the sketch.

For \(f(x)=-(x+1)(x-2)^{2}\), find the \(y\)-intercept, describe the end behavior, and describe the behavior at each zero.

Exercise 13 – Solve a polynomial inequality

Zeros on a number line.

Solve \((x+2)(x-1)(x-4)>0\).

Exercise 14 – Intermediate Value Theorem

Show that \(f(x)=x^{3}-x-1\) has a zero between \(x=1\) and \(x=2\).

Exercise 15 – Evaluate a polynomial

Evaluate \(p(3)\) for \(p(x)=2x^{4}-3x^{3}+x-5\).

Exercise 16 – Box volume

Volume of Priya’s box.

Priya cuts squares with side \(x\) inches from the corners of a \(12\)-by-\(18\)-inch sheet to fold an open box. The volume is \(V(x)=x(12-2x)(18-2x)\). Find \(V(1)\), \(V(2)\) and \(V(3)\), state the realistic domain, and tell which whole-number \(x\) gives the greatest volume.

Exercise 17 – Roller coaster

A hill on Noah’s coaster.

A coaster hill has height \(h(x)=0.01x(x-30)(x-60)\) feet for \(0\le x\le 30\). Find the height at \(x=15\) and at \(x=10\), and explain what the zeros at \(x=0\) and \(x=30\) mean.

Exercise 18 – Zeros with multiplicity

Write a quartic with a double zero at \(1\), simple zeros at \(-2\) and \(3\), and \(f(0)=6\).

Exercise 19 – Least-degree equation

A polynomial of least degree touches the \(x\)-axis at \(-3\), crosses at \(2\), has \(f(0)=-18\) and rises on the right. Write its equation.

Exercise 20 – Find all zeros

Given that \(1\) is a zero of \(f(x)=x^{3}-6x^{2}+11x-6\), find all zeros.

Key facts behind this polynomial graphs worksheet

Key facts

Degree n: at most n real zeros and at most n – 1 turning points.
Even degree: both ends point the same way; odd degree: opposite ways.
Positive leading coefficient: right end rises. Negative: right end falls.
Odd multiplicity crosses the axis; even multiplicity touches it.
f(0) gives the y-intercept; a sign chart solves polynomial inequalities.

How to use this polynomial graphs worksheet

Work through the exercises in order: they go from simple to challenging. Write every step on paper, then compare with the solutions page and redo any exercise you missed the next day.

Download the polynomial graphs worksheet as a PDF with the button below and keep it with your notes.

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