Connect Even Multiplicity with Local Graph Behavior

Supplied materials

• Paper or a browser scratch area; a calculator is optional unless rounding is requested.
• Supplied teaching case: For y=(x−2)^2(x+3), what does the graph do at x=2? Candidate answers: (A) Cannot have value zero there (B) Crosses the x-axis there (C) Has a vertical asymptote there (D) Touches the x-axis and turns without crossing.
• Independent check inputs: For y=(x−3)^3(x+4), what does the graph do at x=3? Candidate answers: (A) Cannot have value zero there (B) Crosses the x-axis there (C) Has a vertical asymptote there (D) Touches the x-axis and turns without crossing.

Learn the idea

Polynomial functions connect symbolic factors with graph features. Division, remainder evaluation, real zeros and multiplicity reveal structure without requiring every coefficient to be expanded, but domains and repeated roots still need attention. A repeated even-power factor stays the same sign on both sides of its zero. If other factors remain nonzero there, the graph touches the axis without crossing locally.

Here is the method on the supplied teaching case. The zero has multiplicity 2. An even factor keeps its sign across the root and touches; an odd factor changes sign and crosses. The other factor stays positive nearby, so this graph touches.

This is local behavior at the named root. Other roots may have different multiplicities and behavior; distinguish the number of distinct zeros from the total degree and the multiplicity of a particular zero.

1. Write or sketch the supplied teaching case, labeling its quantities and restrictions.
2. Apply the method shown in the worked solution, keeping each intermediate equality or justified inference.
3. Solve the independent check with its changed inputs before revealing feedback.
4. Complete the three practice cases; explain which mathematical rule rules out one distractor in each case.

Check your reasoning

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Take it further

Changed-input task: For y=(x−9)^3(x+10), what does the graph do at x=9? Candidate answers: (A) Cannot have value zero there (B) Crosses the x-axis there (C) Has a vertical asymptote there (D) Touches the x-axis and turns without crossing. Show the mathematical reason, not only the selected answer. Expected result: Crosses the x-axis there. The zero has multiplicity 3. An even factor keeps its sign across the root and touches; an odd factor changes sign and crosses. The other factor stays positive nearby, so this graph crosses.

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