Graphs of Polynomials
Use leading terms and multiplicity to sketch polynomial graphs. Includes a worked example, printable practice, and an answer key with explanations.
The answer key follows the practice below. In the editor, remove the final answer-key section before sharing a student copy.
Use This WorksheetIncludes
Standards
Graphs of Polynomials
Name:
Date:
Score:
Learning goal and worked example
Use leading terms and multiplicity to sketch polynomial graphs.
For −(x−1)²(x+2), the graph crosses at −2, turns at 1, rises on the left, and falls on the right. Degree 3 allows at most 2 turning points.
Work with real coefficients. Qualitative sketches show zeros and end behavior; they need not give exact turning-point coordinates.
Read each question carefully and answer to the best of your ability. Show all your work where applicable.
1. Which of the following describes the end behavior of the polynomial function f(x) = -2x^4 + 3x^3 - x + 5?
As x → ∞, f(x) → ∞; As x → -∞, f(x) → ∞
As x → ∞, f(x) → -∞; As x → -∞, f(x) → -∞
As x → ∞, f(x) → ∞; As x → -∞, f(x) → -∞
As x → ∞, f(x) → -∞; As x → -∞, f(x) → ∞
2. What is the maximum number of turning points a polynomial of degree 5 can have?
3
4
5
6
3. If a root of a polynomial has a multiplicity of 2, the graph will the x-axis at that point.
4. The is determined by the highest degree term of the polynomial function.
5. Consider the polynomial function P(x) = (x - 1)^2 (x + 2). What are the zeros of the function and their multiplicities? Describe the behavior of the graph at each zero.
6. Sketch a qualitative graph satisfying the following conditions. Use your own vertical scale and label the zeros and both ends; exact turning-point heights are not required:
- Degree 3
- Leading coefficient is negative
- Zeros at x = -3 (multiplicity 1), x = 0 (multiplicity 1), and x = 2 (multiplicity 1)
7. A polynomial function of odd degree must have at least one real root.
True
False
8. The graph of a polynomial function can have a sharp corner or cusp.
True
False
Answer key and worked solutions
1. b: both ends tend to −∞. 2. b: at most 4 turning points.
3. Touch and turn at. 4. End behavior.
5. x=1 has multiplicity 2: touch and turn. x=−2 has multiplicity 1: cross.
6. One model is −(x+3)x(x−2). It crosses at −3,0,2; the left end rises and the right end falls. Signs are positive, negative, positive, negative in successive intervals. Accept any negative constant multiple and a matching qualitative sketch.
7. True for real-coefficient odd-degree polynomials: opposite end behavior and continuity force a real zero. 8. False: polynomials are differentiable everywhere.
Related Worksheets
Grade 11 Math: Function Characteristics
Identify intercepts, symmetry, domain, range, and end behavior. Includes a worked example, printable practice, and an answer key with explanations.
Transforming Parabolas Worksheet
Use vertex form to describe and graph transformed parabolas. Includes a worked example, printable practice, and an answer key with explanations.
Solving Polynomial Equations Worksheet
Solve polynomial equations using factors, root candidates, and division. Includes a worked example, printable practice, and an answer key with explanations.
Zeros and Multiplicity Worksheet
Relate repeated factors to zeros and local graph behavior. Includes a worked example, printable practice, and an answer key with explanations.
Solving Square Root Equations
Solve radical equations and reject extraneous candidates. Includes a worked example, printable practice, and an answer key with explanations.
Radical Operations Worksheet
Simplify and combine radicals, then rationalize denominators. Includes a worked example, printable practice, and an answer key with explanations.