Graph Factored Polynomials: Intercepts, Multiplicity, and End Behavior

Algebra and Function Practice

Factored form makes key graph features visible. Each factor (x - r) gives an x-intercept at r. The exponent on that factor, its multiplicity, tells whether the graph crosses the axis or turns back. The leading term's degree and sign determine end behavior. In this lesson you will build a sign table, plot intercepts with correct local behavior, and sketch a smooth curve that matches both ends. You will also check your sketch using simple test points and consistency rules. All numerical examples are hypothetical and chosen for easy arithmetic; measure your own paper and tools if you print grids later.

Practice materials: Printable downloads are not available for this lesson yet. Use the instructions and four exercises below.

Read roots and multiplicities from factored form

A polynomial in factored form is a product of linear factors, possibly with exponents. For example, f(x) = (x - 2)^2 (x + 1). Set each factor equal to zero to find roots: x = 2 and x = -1. The exponent 2 on (x - 2) is the multiplicity of root 2; the exponent 1 on (x + 1) is the multiplicity of root -1.

Multiplicity controls local behavior at the intercept. Odd multiplicity means the graph crosses the x-axis. Even multiplicity means the graph touches the axis and turns back. Multiplicity 1 crosses straight through; multiplicity 3 crosses with a flattening near the root; multiplicity 2 touches and turns; multiplicity 4 touches and flattens more. These are qualitative rules, not exact curvature values.

1. Write the polynomial in factored form and list each distinct factor.
2. Set each factor to zero to find the x-intercepts.
3. Record the exponent on each factor as its multiplicity.
4. Label each intercept as cross (odd) or touch (even).
5. Find the y-intercept by evaluating f(0).

Determine end behavior from degree and leading coefficient

The degree is the sum of the multiplicities. The leading coefficient is the product of the coefficients of the highest-degree term in each factor, including any constant multiplier. For f(x) = (x - 2)^2 (x + 1), the degree is 3 and the leading coefficient is 1, so the leading term is x^3. Odd degree with positive leading coefficient means the graph falls to the left and rises to the right.

Use a compact notation: as x approaches negative infinity, write x -> -inf; as x approaches positive infinity, write x -> +inf. For even degree with positive leading coefficient, both ends rise. For even degree with negative leading coefficient, both ends fall. For odd degree with negative leading coefficient, the left end rises and the right end falls.

1. Add the multiplicities to get the degree.
2. Multiply the leading coefficients of the factors, including any outside constant, to get the leading coefficient.
3. Use degree parity and leading coefficient sign to state end behavior.
4. Write end behavior in arrow notation for both directions.
5. Check that the number of turning points is at most degree minus one.

Build a sign table and sketch the graph

A sign table shows whether the polynomial is positive or negative between roots. Choose one test x-value in each interval between consecutive roots, and also outside the smallest and largest roots. Evaluate the sign of each factor, then multiply signs. This tells whether the graph is above or below the x-axis in that interval.

To sketch, draw axes, mark intercepts, and apply end behavior. Between intercepts, draw a smooth curve that stays on the correct side of the axis. At even-multiplicity roots, the curve touches and turns without crossing. At odd-multiplicity roots, it crosses. Use the y-intercept and one or two test points to set approximate height. Keep the curve smooth; avoid sharp corners.

1. List roots in increasing order and mark intervals.
2. Pick a test value in each interval and record the sign of f.
3. Plot x-intercepts and the y-intercept.
4. Draw end behavior arrows first.
5. Connect through intercepts with correct cross or touch behavior and correct sign in each interval.
6. Verify with one additional test point.

Check your sketch for consistency

A correct sketch must satisfy several checks at once. The number of x-intercepts matches the distinct real roots. Each intercept's cross or touch behavior matches its multiplicity parity. The sign table matches whether the curve is above or below the axis. End behavior matches degree and leading coefficient. The y-intercept matches f(0). The curve is smooth and has no more turning points than degree minus one.

If any check fails, revise the sketch rather than forcing the curve. Common fixes include swapping a touch for a cross, reversing one end, or placing a test point on the wrong side. Because sketches are qualitative, small height differences are acceptable if all structural checks pass.

1. Count distinct roots and compare with plotted intercepts.
2. Check parity at each intercept.
3. Compare each interval sign with the curve's position.
4. Confirm both end arrows.
5. Confirm y-intercept and turning-point count.
6. Mark the sketch as consistent or revise.

Worked example

Sketch f(x) = -(x + 2)(x - 1)^2. Identify roots, multiplicities, end behavior, y-intercept, and interval signs.

Four practice exercises

Common mistakes

• Treating every x-intercept as a straight cross. Check the exponent on each factor. Even multiplicity means touch and turn; odd means cross.
• Using the number of factors instead of the sum of multiplicities for degree. Add all exponents, including repeated factors, to get the degree.
• Ignoring a negative constant multiplier when stating end behavior. Multiply the leading coefficient signs of all factors and any outside constant before deciding end behavior.
• Placing the curve on the wrong side of the axis between roots. Evaluate one test value per interval and record the sign before sketching.
• Drawing sharp corners at intercepts. Keep the curve smooth; at a touch point, approach and leave on the same side without a corner.

Self-check rubric

• All distinct real roots are plotted at correct x-values.
• Each intercept is labeled cross or touch according to multiplicity parity.
• End behavior matches degree parity and leading coefficient sign.
• Sign table intervals match the curve's position above or below the axis.
• y-intercept is correct and plotted.
• Curve is smooth with no more than degree minus one turning points.
• Valid alternative sketches with different heights but same structural features are accepted.

Questions and answers