Precalculus Polynomial Functions and Their Graphs Test with Point Values (PDF)

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

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This polynomial graphs test 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.

This 20-point test covers degree, end behavior, zeros and multiplicity, graphs, inequalities and a volume model (F-IF.C.7c, A-APR.B.3). You have 45 minutes. Show your work.

Time and points

About this test

Grade: Precalculus
Chapter: Polynomial Functions and Their Graphs
Standards: F-IF.C.7c, A-APR.B.3, F-IF.B.4
Time: 45 minutes
Total: 20 points

Polynomial graphs test: questions

Question 1 – Degree and end behavior [2 pts]

For \(f(x)=-5x^{3}+x^{2}-7\), state the degree, the leading coefficient and the end behavior.

Question 2 – Zeros and multiplicity [2 pts]

Graph of f.

For \(f(x)=(x+2)^{2}(x-5)\), list each zero with its multiplicity and tell whether the graph crosses or touches the \(x\)-axis there.

Question 3 – Factor and solve [2 pts]

Find all zeros of \(f(x)=x^{3}-4x^{2}-9x+36\).

Question 4 – Write a polynomial [2 pts]

Write a polynomial with leading coefficient 1 and zeros \(-2\), \(1\) and \(3\) in standard form.

Question 5 – Read a graph [3 pts]

Graph of a cubic.

(a) State the zeros. (b) State the least possible degree and the sign of the leading coefficient. (c) Write the equation.

Question 6 – Polynomial inequality [2 pts]

Solve \((x+1)(x-3)(x-5)\le 0\).

Question 7 – Intermediate Value Theorem [2 pts]

Show that \(f(x)=x^{3}+2x-5\) has a real zero between 1 and 2.

Question 8 – Box volume [3 pts]

Volume of an open box.

Ethan cuts squares of side \(x\) inches from a \(10\)-by-\(16\)-inch sheet. The volume is \(V(x)=x(10-2x)(16-2x)\). (a) Find \(V(1)\) and \(V(2)\). (b) State the domain. (c) Find \(V(3)\) and tell which of \(x=1,2,3\) gives the greatest volume.

Question 9 – Multiplicity and degree [2 pts]

For \(f(x)=x^{2}(x-3)^{3}(x+1)\), state the degree and the behavior of the graph at each zero.

Key facts behind this polynomial graphs test

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 test

Time yourself: the test is designed for an average student in 45 minutes. Answer the questions you are sure of first, then come back to the longer ones and check your units.

All Precalculus chapters: Precalculus math.

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