S. D. V. Stephens

notes / Integrated Mathematics / chapter 10: Polynomials

10.3 Sketching Polynomials and Finding Them from Their Zeros

Putting the degree, the end behavior and the multiplicities together gives a sketch of a polynomial from its factored form. Running the same facts backwards recovers a polynomial from its zeros or from its graph.

Sketching

To sketch a polynomial: (1) the leading term, which gives the degree, the max number of turning points and the end behavior; (2) the -intercept; (3) the -intercepts with their multiplicities.

Example 10.3.1. .

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The zeros are , , , and (not real numbers). The exponents add up to (the contributes ), so the end behavior is .

  1. Leading term: .
    • Degree: .
    • Max turning points: .
    • End behavior: like : down on both ends.
  2. -intercept: . Remember: this is obtained via , that is,
  3. -intercepts: with multiplicity , so it crosses; with multiplicity , so it bounces; with multiplicity , so it crosses.

Note. Max turning points , where is the leading term.

Example 10.3.2. Sketch .

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  1. Leading term .
    • Degree: .
    • Max turning points: .
    • End behavior: down on the left, up on the right.
  2. -intercept: .
  3. -intercepts: with multiplicity , so it bounces; with multiplicity , so it crosses.

Two more points: and .

Graph of f(x)=x^2(x-3): it touches the x-axis at the origin, dips through (1,-2) and (2,-4), and crosses the x-axis at 3.

Example 10.3.3. Sketch .

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The leading term is .

  1. Leading term: .
    • Degree: .
    • Max turning points: .
    • End behavior: down on both ends.
  2. -intercept: .
  3. -intercepts: with multiplicity , so it crosses; with multiplicity , so it crosses (flattening as it does).
Graph of f(x)=-2(x+2)(x-2)^3: it crosses the x-axis at -2, peaks near x=-1, passes through (0,32), flattens as it crosses at 2, and falls on both ends.

Example 10.3.4. Sketch .

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Rewrite: .

  1. Leading term: .
    • Degree: .
    • Max turning points: .
    • End behavior: up on the left, down on the right.
  2. -intercept: .
  3. -intercepts: and . The zero has multiplicity , so it bounces; has multiplicity , so it crosses.
Graph of h(x)=-(x+4)^2(x-1): it touches the x-axis at -4, rises through (0,16), then crosses the x-axis at 1 and falls.

Example 10.3.5. .

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Factor: .

  1. Leading term .
    • Degree: .
    • Max turning points: .
    • End behavior: down on both ends.
  2. -intercept: .
  3. -intercepts: , , , , each with multiplicity , so the graph crosses at all four.

Example 10.3.6. .

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Factor: .

  1. Leading term .
    • Degree: .
    • Max turning points: .
    • End behavior: up on both ends.
  2. -intercept: , since .
  3. -intercepts: with multiplicity , so it bounces; and with multiplicity , so it crosses.

Functions from -intercepts

Example 10.3.7. .

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. X: top , bottom , sides and . So

Going the other way: suppose we know the -intercepts and want the function.

Example 10.3.8. Suppose the -intercepts are , , and .

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So , and

Example 10.3.9. Zeros: ; degree .

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So , and

Example 10.3.10. Zeros: with multiplicity , with multiplicity ; degree .

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and , used twice: . So

Example 10.3.11. Zeros: with multiplicity , with multiplicity , with multiplicity , with multiplicity ; degree .

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so The factor can also be written : since , the two give the same zero with the same multiplicity, and only the constant changes.

Functions from graphs

Example 10.3.12. An odd-degree graph crosses the -axis at , and . Find a function for it.

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A cubic-shaped graph rising from the lower left, crossing the x-axis at 0, 1 and 2, and rising to the upper right.

so

Example 10.3.13. An even-degree graph crosses the -axis at , bounces off it at , crosses at , and passes through . Find a function for it.

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A graph rising on both ends that crosses the x-axis at -4, touches it and turns back at -1, crosses at 3, and passes through (1,-8).

At it crosses, so that zero has odd multiplicity; at it bounces, so that zero has even multiplicity; at it crosses, so odd again. There are zeros, but the bounce counts twice: , thus the degree is at least . With multiplicities , , : . Using the point : So

Example 10.3.14. An odd-degree graph crosses the -axis at , and , and passes through . Find a function for it.

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A cubic-shaped graph falling from the upper left, crossing the x-axis at -1, passing through (0,-1), crossing at 1 and 2, and falling to the lower right.

The zeros are , , , so Let’s say the point is , i.e. : So

Two problems on what a graph says about its polynomial; only the answers are in the notes.

Problem 10.3.15.

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It means is of degree , not necessarily that .

Problem 10.3.16.

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(a) There are 5 changes in direction.

(b) As , , because (which grows the fastest; shrinks, I guess?) is such that . If even, it is such that…

At , :

Exercises

Exercise 10.3.17.

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At and (unneeded).

Graph of the cubic f(x) = 0.2(x-3)(x+2)(x+3), crossing the x-axis at -3, -2 and 3, with f(0) = -3.6 and f(1) = -4.8 marked.

Yay!