A polynomial graph turns algebraic structure into shape. Factors identify horizontal-axis intersections, multiplicity affects how the graph behaves there, and the leading term controls the distant ends of the graph.
1. Roots become horizontal-axis intersections
If P(x)=(x−2)(x+1), the roots are 2 and −1. Therefore y=P(x) crosses or touches the horizontal axis at x=2 and x=−1.
2. Odd multiplicity changes sign
Near a simple root such as x=2 in (x−2)(x+1), the factor x−2 changes sign as x passes through 2. If the other factors remain nonzero, the polynomial changes sign, so the graph crosses the axis.
3. Even multiplicity need not change sign
For P(x)=(x−3)²(x+1), the squared factor is nonnegative on both sides of x=3. The graph touches the horizontal axis at x=3 and turns back rather than crossing there.
At x=−1, the multiplicity is 1, so the graph crosses.
4. End behaviour comes from the leading term
For large |x|, lower-degree terms become relatively less important than the leading term.
- Positive even degree: both ends rise.
- Negative even degree: both ends fall.
- Positive odd degree: left falls, right rises.
- Negative odd degree: left rises, right falls.
5. Worked sketch from factor form
Sketch the key structure of y=(x+2)(x−1)².
It is a cubic with positive leading coefficient, so the left end falls and right end rises. Roots are −2 with multiplicity 1 and 1 with multiplicity 2. Therefore the graph crosses at −2 and touches at 1.
The vertical-axis intercept is y(0)=2. These features provide enough anchors for a qualitative sketch.
6. Sign chart as a graph check
For (x+2)(x−1)², choose test values. At x=−3 the polynomial is negative. At x=0 it is positive. At x=2 it remains positive. Thus the sign changes at −2 but not at 1, agreeing with the multiplicities.
7. Degree limits turning points
A polynomial of degree n can have at most n−1 turning points. A cubic can therefore have at most two. This is a useful plausibility check on a sketch.
The statement is an upper bound, not a promise. A cubic such as y=x³ has no turning point.
8. Expanded form can hide graph information
The cubic x³−3x+2 factorises as (x−1)²(x+2). The expanded form shows coefficients; the factorised form immediately shows a repeated root at 1 and another root at −2.
Switch representation before sketching when factorisation is available.
9. Common mistakes
- Drawing every root as a crossing.
- Using the constant term to decide end behaviour.
- Drawing more turning points than the degree permits.
- Ignoring the vertical-axis intercept when checking a sketch.
10. Practice
- State the end behaviour of y=2x⁴−x+1.
- State the end behaviour of y=−x³+4x.
- For y=(x−2)²(x+3), identify each root and whether the graph crosses or touches.
- Find the vertical-axis intercept of y=(x+1)(x−4)(x−2).
- What is the maximum possible number of turning points of a degree-5 polynomial?
11. Answers
- Both ends rise.
- Left rises; right falls.
- x=2 multiplicity 2: touches; x=−3 multiplicity 1: crosses.
- y(0)=(1)(−4)(−2)=8.
- 4.
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