Precalculus Polynomial Functions and Their Graphs Worksheet Solutions (PDF)
This polynomial graphs answers 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.
Precalculus · Common Core State Standards: F-IF.C.7c, A-APR.B.3, F-IF.B.4
Polynomial graphs answers: exercises 1 to 10
Exercise 1 – Degree and leading coefficient
The highest power is \(x^{3}\) with coefficient 5.
Degree 3; leading coefficient 5
Exercise 2 – Expand a product
\((x-2)(x+3)=x^{2}+x-6\). Multiply by \((x+1)\): \(x^{3}+2x^{2}-5x-6\).
\(x^{3}+2x^{2}-5x-6\)
Exercise 3 – End behavior, odd degree
Odd degree with a negative leading coefficient: left end up, right end down.
As \(x\to -\infty\), \(f(x)\to\infty\); as \(x\to\infty\), \(f(x)\to-\infty\)
Exercise 4 – End behavior, even degree
Even degree with a positive leading coefficient: both ends up.
\(f(x)\to\infty\) as \(x\to\pm\infty\)
Exercise 5 – Zeros from factors
Set each factor equal to 0.
\(x=0,\ 4,\ -5\)
Exercise 6 – Multiplicity
\(x=1\) has multiplicity 2 (even): touches and turns. \(x=-3\) has multiplicity 3 (odd): crosses with a flattening. Degree \(=2+3=5\).
\(1\) (mult. 2, touch); \(-3\) (mult. 3, cross); degree 5
Exercise 7 – Factor out x
\(x^{3}-9x=x(x^{2}-9)=x(x-3)(x+3)\).
\(x=0,\ 3,\ -3\)
Exercise 8 – Factor by grouping
\(x^{2}(x+2)-9(x+2)=(x+2)(x^{2}-9)=(x+2)(x-3)(x+3)\).
\(x=-2,\ 3,\ -3\)
Exercise 9 – Build a polynomial
\(f(x)=a(x+1)(x-2)(x-4)\). \(f(0)=a(1)(-2)(-4)=8a=16\), so \(a=2\). Expand: \(2x^{3}-10x^{2}+4x+16\).
\(f(x)=2(x+1)(x-2)(x-4)=2x^{3}-10x^{2}+4x+16\)
Exercise 10 – Read a graph
Crossing at \(-3\): factor \((x+3)\). Touching at 1: factor \((x-1)^{2}\). The right end rises, so \(a>0\). With \(y\)-intercept \(3\): \(a(3)(1)=3\), \(a=1\).
\(f(x)=(x+3)(x-1)^{2}\)
Exercises 11 and up
Exercise 11 – Turning points
Turning points are at most \(n-1\), so a degree-6 polynomial has at most 5. With 3 turning points, \(n-1\ge 3\), so \(n\ge 4\).
(a) 5; (b) 4
Exercise 12 – Sketch a polynomial
\(f(0)=-(1)(4)=-4\). Degree 3 with a negative leading coefficient: left up, right down. At \(-1\) the graph crosses; at \(2\) it touches.
\(y\)-int \(-4\); left up, right down; cross at \(-1\), touch at \(2\)
Exercise 13 – Solve a polynomial inequality
Test values: \(x=-3\): \((-1)(-4)(-7)< 0\). \(x=0\): \((2)(-1)(-4)=8>0\). \(x=2\): \((4)(1)(-2)< 0\). \(x=5\): \((7)(4)(1)>0\).
\((-2,1)\cup(4,\infty)\)
Exercise 14 – Intermediate Value Theorem
\(f(1)=1-1-1=-1< 0\) and \(f(2)=8-2-1=5>0\). A polynomial is continuous, so it must cross 0 between 1 and 2.
A zero exists in \((1,2)\)
Exercise 15 – Evaluate a polynomial
\(2(81)-3(27)+3-5=162-81+3-5\).
79
Exercise 16 – Box volume
\(V(1)=1(10)(16)=160\). \(V(2)=2(8)(14)=224\). \(V(3)=3(6)(12)=216\). The sides \(12-2x\) and \(18-2x\) must be positive, so \(0< x< 6\).
\(160,\ 224,\ 216\) in\(^{3}\); \(0< x< 6\); \(x=2\)
Exercise 17 – Roller coaster
\(h(15)=0.01(15)(-15)(-45)=101.25\). \(h(10)=0.01(10)(-20)(-50)=100\). The zeros are where the track is at ground level, the start and end of the hill.
\(101.25\) ft; \(100\) ft; ground level at both ends
Exercise 18 – Zeros with multiplicity
\(f(x)=a(x-1)^{2}(x+2)(x-3)\). \(f(0)=a(1)(2)(-3)=-6a=6\), so \(a=-1\). Expanded: \(-x^{4}+3x^{3}+3x^{2}-11x+6\).
\(f(x)=-(x-1)^{2}(x+2)(x-3)=-x^{4}+3x^{3}+3x^{2}-11x+6\)
Exercise 19 – Least-degree equation
\(f(x)=a(x+3)^{2}(x-2)\). \(f(0)=a(9)(-2)=-18a=-18\), so \(a=1\). Degree 3 with \(a>0\) rises on the right.
\(f(x)=(x+3)^{2}(x-2)\)
Exercise 20 – Find all zeros
\((x-1)\) is a factor: \(x^{3}-6x^{2}+11x-6=(x-1)(x^{2}-5x+6)=(x-1)(x-2)(x-3)\).
\(x=1,\ 2,\ 3\)
Key facts behind this polynomial graphs answers
- 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 answers
Cover each solution, solve the exercise again, then uncover it and compare every line. If your answer differs, find the first step that changed and redo the exercise from there.
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