Lesson 3 of 3
Combine intercepts, signs, ends, and checking points to sketch a polynomial.
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Worked example
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For , label x-intercepts and . The y-intercept is .
Big ideas
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Start with degree and leading coefficient for end behavior. Factor where possible, then list each real zero and its multiplicity.
The x-intercepts are at real zeros r. The y-intercept is . An input zero is different from a zero output.
Choose a test input in each interval between consecutive zeros. Use factor signs to tell whether the graph is above or below the axis. Add extra points where helpful.
Polynomials are continuous and smooth. Draw no holes, jumps, or sharp corners. A rough sketch from factors need not locate exact turning points; label only verified coordinates.
Worked example
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Sketch roughly. It is cubic with negative leading coefficient, so left end up and right end down. It crosses at , touches at 1, and passes through . Test inputs and 2 give 16 and . Join smoothly; exact turning points are not established by this information.
Step-through
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Sketch the main features of .
Whole picture, 4 steps
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Worked example
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A cubic has simple zeros -1, 2, and 4 and passes through . Write . Then , so . Its left end falls and right end rises.
Worked example
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For , the zeros divide the line into three intervals. At the value is -12; at it is 12; at it is -12. Thus the graph is below, above, then below the axis, consistent with its negative leading quadratic term.
Mix-up alert
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Wrong idea: draw a pointed corner at a repeated polynomial zero. Repair: the polynomial graph stays smooth. At an even-multiplicity zero it touches and turns back without a sharp corner.
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Flashcards
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Look at the highest-power term.
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What is the first useful information for end behavior?
The degree and leading coefficient.
How do you find the y-intercept?
Evaluate f(0) and write .
What points come from zeros -2 and 5?
and .
What do interval test points tell you?
Whether the graph is above or below the x-axis there.
Can a polynomial graph have a hole?
No.
Can a rough factor sketch prove an exact local maximum coordinate?
No; that coordinate needs additional calculation or evidence.
What formula starts a quadratic with zeros 1 and 6?
p(x)=a(x-1)(x-6), where a is nonzero.
How can you find the missing nonzero scale a?
Use another known point that is not one of the zeros.
True or false
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Self-test
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Self-test
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6 questions. One at a time.
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