A nonconstant polynomial function of x is any function representation that is equivalent to the analytical form below. The exponents are whole numbers counting down to 0, the coefficients ai are real numbers, n is a positive integer, and the first coefficient an is not zero.
| Name | What it is | In 2x⁴ − 5x + 1 |
|---|---|---|
| Degree | The highest exponent, n | 4 |
| Leading term | anxn | 2x⁴ |
| Leading coefficient | an, the number multiplying the leading term | 2 |
Pick an expression. The tool decides whether it is a nonconstant polynomial function, and for the ones that are, it reads off the degree, the leading term, and the leading coefficient.
Every exponent is a positive whole number (3 and 1), the constant term 7 is a real number, and the coefficients are real.
| degree n | leading term | leading coefficient |
| 3 | 4x³ | 4 |
Where a polynomial function switches between increasing and decreasing, it has a local (or relative) maximum or minimum output value. Local extrema also occur at the included endpoint of a polynomial with a restricted domain.
A local maximum is a global (or absolute) maximum when it is greater than all other output values. A local minimum is a global minimum when it is less than all other output values.
The curve is green where the outputs are increasing and coral where they are decreasing. Every dot is a local maximum or minimum; the gold dots are also global, and the silver dots are local only.
| point | type | global? |
|---|---|---|
| (−1, 2) | local maximum | local only |
| (1, −2) | local minimum | local only |
Between every two distinct real zeros of a nonconstant polynomial function, there must be at least one input value corresponding to a local maximum or local minimum.
Here is why in plain words. At one zero the output is 0. Between the zeros the graph is either all above the x-axis or all below it, and it must come back to 0 at the next zero. So it has to leave the axis, turn around, and return. The turn is the local maximum or minimum. The average rate just after the left zero and the average rate just before the right zero have opposite signs.
Choose a polynomial, then choose two neighboring real zeros. The gold stretch of the graph sits between them. Look for the local maximum or minimum hiding in that stretch, and check the average rates just inside each zero.
A polynomial function of even degree has either a global maximum or a global minimum. With a positive leading coefficient it opens upward and has a global minimum. With a negative leading coefficient it opens downward and has a global maximum. A polynomial of odd degree has neither, because its outputs keep going up on one side and down on the other.
Read the degree and the sign of the leading coefficient, then see what kind of global extreme each polynomial has. Gold dots mark the global extremum, silver dots are other local extrema.
For a quadratic function, the global maximum or global minimum occurs at the vertex. If a > 0 the parabola opens up and the vertex is the global minimum. If a < 0 it opens down and the vertex is the global maximum. From f(x) = ax² + bx + c, the vertex is at x = −b/(2a); from vertex form a(x − h)² + k, it is at (h, k).
A quadratic in vertex form is f(x) = a(x − h)² + k. Slide a, h, and k. The vertex is the single point where the function switches between increasing and decreasing, and it is the global maximum or minimum.
Points of inflection of a polynomial function occur at input values where the rate of change of the function changes from increasing to decreasing or from decreasing to increasing. This is where the graph changes from concave up to concave down, or from concave down to concave up.
You already know how to read this from Lesson 1.3. List the average rates over equal-length intervals. If they were increasing and then start decreasing, the graph was concave up and then becomes concave down, and an inflection point sits where it switches.
Chords are green where the average rates are increasing and coral where they are decreasing, as in Lesson 1.3. A point of inflection is where the colors switch. Turn on the extrema to see that inflection points and local maximums and minimums are different things.
| interval | average rate | compared with the previous |
|---|---|---|
| [−3, −2.5] | 19.75 | — |
| [−2.5, −2] | 12.25 | ▼ decreasing |
| [−2, −1.5] | 6.25 | ▼ decreasing |
| [−1.5, −1] | 1.75 | ▼ decreasing |
| [−1, −0.5] | -1.25 | ▼ decreasing |
| [−0.5, 0] | -2.75 | ▼ decreasing |
| [0, 0.5] | -2.75 | = same |
| [0.5, 1] | -1.25 | ▲ increasing |
| [1, 1.5] | 1.75 | ▲ increasing |
| [1.5, 2] | 6.25 | ▲ increasing |
| [2, 2.5] | 12.25 | ▲ increasing |
| [2.5, 3] | 19.75 | ▲ increasing |
Every characteristic in this lesson connects back to rates of change. Use the table as a map from what you see to what it means.
| You see | It means | Rates of change |
|---|---|---|
| Local maximum | Outputs switch from increasing to decreasing | Rates go from positive to negative |
| Local minimum | Outputs switch from decreasing to increasing | Rates go from negative to positive |
| Two distinct real zeros | At least one local extremum between them | Rates just inside the zeros have opposite signs |
| Even degree | A global maximum or a global minimum, by the sign of the leading coefficient | Quadratic: switch occurs at the vertex |
| Concavity switches | A point of inflection | Rates go from increasing to decreasing, or the reverse |
Study the graph and try to predict each row of the report card. Click a row to check yourself. The whole page comes together here: degree, zeros, extrema, increasing and decreasing, and concavity.