Graphs of Polynomial Functions
The graph of a polynomial function is a smooth, continuous curve whose end behavior is determined by the degree and the sign of the leading coefficient, and whose x-intercepts are the real zeros. In brief: a polynomial of degree n has at most n real zeros and at most n − 1 turning points. For f(x) = x3 − 4x the graph falls on the left, rises on the right, crosses the x-axis at −2, 0 and 2, and has two turning points.
You can sketch a polynomial graph without a table of many values: find the end behavior, the intercepts, the multiplicity of each zero and a few test points. This guide shows how, explains how multiplicity changes the shape at an intercept and provides a table of the possible end behaviors.
What is the end behavior of a polynomial graph?
For large |x|, the term with the highest power dominates, so the graph behaves like anxn. Two things decide the shape at the ends:
- Even degree: both ends go in the same direction. Up on both sides if the leading coefficient is positive, and down on both sides if it is negative (like parabolas).
- Odd degree: the ends go in opposite directions. For a positive leading coefficient the graph falls on the left and rises on the right; for a negative one it rises on the left and falls on the right (like y = x3).
Example: f(x) = −2x4 + x2 has an even degree and a negative leading coefficient, so f(x) → −∞ as x → ±∞. For g(x) = x5 − 3x, g(x) → +∞ as x → ∞ and −∞ as x → −∞.
How does the multiplicity of a zero change the graph?
If a factor (x − r) appears k times in the factorization, r is a zero of multiplicity k. Near the zero, the graph behaves like (x − r)k.
- k = 1: the graph crosses the x-axis in a nearly straight way.
- k even (2, 4, …): the graph touches the axis and turns back, without crossing, flattened at the zero for k = 4.
- k odd and greater than 1 (3, 5, …): the graph crosses the axis and flattens at the zero, like y = x3 at the origin.
The total of the multiplicities of the real zeros is at most the degree.
How do you sketch a polynomial graph step by step?
- Determine the end behavior from the degree and the leading coefficient.
- Find the y-intercept f(0).
- Factor the polynomial to find the x-intercepts and their multiplicities.
- Mark the intercepts and use test points between them to find the sign.
- Draw a smooth curve through the points, crossing or touching according to the multiplicities.
- Check that the number of turning points does not exceed degree − 1.
Worked example with a figure
Sketch f(x) = (x + 2)(x − 1)2(x − 3).
The degree is 4 with a positive leading coefficient, so both ends go up. The y-intercept is f(0) = 2 · 1 · (−3) = −6. The zeros are x = −2 (multiplicity 1), x = 1 (multiplicity 2) and x = 3 (multiplicity 1). So the graph crosses at −2 and 3 and touches the axis at 1.
Signs: for x < −2 the function is positive (the ends go up). Between −2 and 1 it is negative, and between 1 and 3 it is also negative, because the double zero does not change the sign. For x > 3 it is positive. Test: f(2) = 4 · 1 · (−1) = −4 ✓.
The sketch shows the main features: the shape of the ends, the intercepts and the behavior at the double zero. An exact plot needs a table of values.
End behavior table
| Degree | Leading coefficient | Left end | Right end | Example |
|---|---|---|---|---|
| Even | Positive | up | up | x2 |
| Even | Negative | down | down | −x4 |
| Odd | Positive | down | up | x3 |
| Odd | Negative | up | down | −x5 |
Where do these ideas come from?
Graphs of polynomials became a subject of study after the invention of coordinate geometry. Isaac Newton classified cubic curves in the seventeenth century. The fact that a polynomial of degree n has at most n real zeros is a consequence of the factor theorem, and the results about turning points were put on a firm footing with calculus.
Where are polynomial graphs used today?
Polynomial models describe the trajectory of projectiles (degree 2), the volume of a box as a function of a cut-out size (degree 3) and fitted curves in statistics. Designers use cubic polynomials for smooth curves in fonts and animation paths. Reading the shape of a graph helps to decide how many parameters a model needs.
Which mistakes are common?
Making the graph cross at every zero
At zeros of even multiplicity the graph touches and turns around.
Drawing corners
Polynomial graphs are smooth. They have no sharp corners or breaks.
Wrong end behavior
Look only at the highest power. In −x3 + 100x2 the cubic term wins for large |x|.
Too many turning points
A degree 3 polynomial has at most 2 turning points. More means a mistake in the sketch.