Polynomial Functions Worksheet: Equations, Tables & Graphs
Connect polynomial equations, tables, and graphs with this free Algebra 2 worksheet. Complete eight multiple-representation organizers covering standard form, factored form, zeros, multiplicity, and end behavior. Download the student worksheet, worked answer key, reference organizer, or complete bundle below.
Download the worksheet and answers
Student worksheet: 10 pages · Teacher key: 10 pages · US Letter PDFs
The student PDF includes eight organizers and a reference page. The teacher key fills in every box and includes graphs, teaching notes, and scoring guidance.
Inside the student worksheet
Function A begins with these two forms:
f(x) = x2 + x − 2
f(x) = (x + 2)(x − 1)
| x | −3 | −2 | 0 | 1 | 2 |
|---|---|---|---|---|---|
| f(x) | ___ | ___ | ___ | ___ | ___ |
Worked example: connect an equation, table, and graph
Start with m(x) = (x + 2)2(x − 1). This sample is separate from the eight functions in the printable worksheet.
Read the factored form
The factor (x + 2)2 gives a zero at −2 with multiplicity 2. The graph touches the x-axis there. The factor (x − 1) gives a zero at 1 with multiplicity 1, where the graph crosses.
Expand to standard form
m(x) = (x2 + 4x + 4)(x − 1)
m(x) = x3 + 3x2 − 4
The degree is 3 and the leading coefficient is positive, so the left end falls and the right end rises.
Make a table
Substitute each x-value into either equation form. For example, m(−3) = (−1)2(−4) = −4.
| x | −3 | −2 | 0 | 1 |
|---|---|---|---|---|
| m(x) | −4 | 0 | −4 | 0 |
At x = 0, the output is −4. Both equation forms must therefore give the same y-intercept, (0, −4).
Sketch the graph and describe it
Plot the table values and mark the two zeros. Draw a smooth curve that touches at −2, passes through (0, −4), and crosses at 1. Its left end points down; its right end points up.
A description for the organizer could be: “The graph touches the x-axis at −2, crosses at 1, and passes through (0, −4).”
Check one more point. Choose x = −1, which is not in the table above. The factored form gives (1)2(−2) = −2. The standard form gives −1 + 3 − 4 = −2. The graph should pass through (−1, −2).
Write a polynomial equation from graph clues
Now work in the other direction. Suppose a polynomial has degree 3, touches the x-axis at 2, crosses at −1, and passes through (0, −8).
- Write the factors. Touching at 2 needs an even multiplicity; crossing at −1 needs an odd multiplicity. The stated degree of 3 fixes those multiplicities at 2 and 1.
- Include the coefficient. Write p(x) = a(x − 2)2(x + 1).
- Use the point. Substituting (0, −8) gives −8 = a(−2)2(1) = 4a, so a = −2.
p(x) = −2(x − 2)2(x + 1)
The negative leading coefficient gives a left end that rises and a right end that falls. The degree matters here: it rules out extra factors. The point fixes the leading coefficient.
View the Six Ways reference organizer
Keep this beside the worksheet to see how the equation forms, table, graph, zeros, and end behavior fit together.
Choosing pages for your class or tutoring session
Students need some practice with factoring, expanding, reading intercepts, and odd and even multiplicity before starting. Choose the pages that fit the skill you want them to work on.
| Level | Pages | Focus | Time |
|---|---|---|---|
| Guided | A–B | Quadratic and cubic functions with more information supplied. | 20–25 minutes |
| Standard | C–E | Cubic and quartic functions; connect factors, intercepts, and scale. | 35–45 minutes |
| Challenge | F–H | Repeated roots, fractional coefficients, and fewer starting clues. | 40–55 minutes |
For partner work, have one student complete an equation form and the other complete the table or graph. Then compare the results using a point they calculate together.
For independent practice, require the extra-point check before students turn in an organizer. Correct zeros can still lead to a graph drawn at the wrong scale.
Mistakes to check
- Reversing the zero’s sign. A factor of x + 2 gives x = −2. Write x + 2 = 0 first.
- Treating every zero as a crossing. Even multiplicity touches and turns. Odd multiplicity crosses, including a triple root that flattens near the axis.
- Leaving the coefficient at 1. End behavior tells you the sign of the leading coefficient. Use a supplied point to find its size.
If students need a review, use the factoring polynomials guide or the graphing polynomial functions worksheet. The polynomial graph card sort offers matching practice before students complete missing representations on their own.
Answer key preview
These are the first three functions in the worksheet. The full teacher key includes each completed organizer, its graph, and a note explaining why the supplied clues determine the answer.
| Function | Standard form | Factored form | Zeros and multiplicity | End behavior |
|---|---|---|---|---|
| A | x2 + x − 2 | (x + 2)(x − 1) | −2 and 1; both multiplicity 1 | Both ends up |
| B | −x3 + 3x + 2 | −(x + 1)2(x − 2) | −1: multiplicity 2; 2: multiplicity 1 | Left up; right down |
| C | 2x3 − 6x + 4 | 2 · (x + 2)(x − 1)2 | −2: multiplicity 1; 1: multiplicity 2 | Left down; right up |
Questions about the worksheet
What are multiple representations of a polynomial function?
They are different ways to show the same function: equations, a table, a graph, and a description. This worksheet also asks students to record zeros, multiplicities, and end behavior so they can check that the representations agree.
Do students need a graphing calculator?
No. The activity can be completed by hand. A calculator can help check arithmetic or compare a finished sketch with the function’s graph.
Can I share the PDFs with students and colleagues?
Yes. You may print them, share them with your department, or post them in your class LMS. For a public resource page, link to this page so readers can find the worksheet and its answer key together.
More polynomial practice
For another explanation of repeated roots, see Illustrative Mathematics: Multiplicity or MathBitsNotebook: Zeros and Multiplicity.