Algebra 101 - Unit 7

Lesson 41: Polynomial Functions and Graphs (End Behavior, Turning Points)

Polynomial functions are expressions of the form f(x) = aₙxⁿ + aₙ₋₁xⁿ⁻¹ + ... + a₁x + a₀. Graphing polynomials involves understanding their degree, leading coefficient, end behavior, and turning points.

End Behavior

End behavior depends on the degree (n) and leading coefficient (aâ‚™):

• Degree even, leading coefficient positive → both ends up
• Degree even, leading coefficient negative → both ends down
• Degree odd, leading coefficient positive → left down, right up
• Degree odd, leading coefficient negative → left up, right down

Example: f(x) = 2x³ − x² + 3 → degree 3 (odd), leading coefficient 2 (positive) → left down, right up

Turning Points

The maximum number of turning points of a polynomial function is n − 1, where n is the degree.

Example: f(x) = x⁴ − 4x² → degree 4 → up to 3 turning points

Graphing Steps

1. Identify the degree and leading coefficient → determine end behavior
2. Find zeros (roots) of the polynomial
3. Determine multiplicity of each zero → affects whether graph crosses or touches x-axis
4. Plot turning points and intercepts
5. Sketch smooth curve connecting points respecting end behavior

Examples

Example 1: f(x) = x³ − 3x²

• Degree 3 (odd), leading coefficient 1 (positive) → left down, right up
• Factor: x²(x − 3) → zeros at x = 0 (multiplicity 2), x = 3 (multiplicity 1)
• Zero with even multiplicity → touches x-axis, zero with odd multiplicity → crosses

Example 2: f(x) = −2x⁴ + 4x²

• Degree 4 (even), leading coefficient −2 (negative) → both ends down
• Factor: −2x²(x² − 2) → zeros at x = 0, x = ±√2
• Plot turning points → sketch smooth curve

Practice Problems

1. Determine end behavior and number of turning points: f(x) = x⁵ − 2x³ + x
2. Factor and graph: f(x) = x³ − 4x
3. Find zeros and end behavior: f(x) = −x⁴ + 2x²
4. Describe turning points for f(x) = x⁴ − 8x² + 12
5. Sketch graph: f(x) = 2x³ − 3x² − 12x + 5