Section 3.2: End Behavior & Graphs
This section covers the end behavior of polynomials based on degree and leading coefficient and basic sketching of polynomial graphs.
Example 1: Determine End Behavior
Find the end behavior of \( f(x) = 2x^4 - x^3 + 5 \).
Step 1: Degree = 4 (even), leading coefficient = 2 (positive)
Step 2: End behavior: as x → ±∞, f(x) → +∞ → graph rises on both ends
Example 2: Sketch a Cubic Polynomial
Sketch \( g(x) = -x^3 + 3x^2 - x \).
Step 1: Degree = 3 (odd), leading coefficient = -1 (negative)
Step 2: End behavior: falls left, rises right → graph falls left, rises right
Step 3: Find zeros: factor \( g(x) = -x(x^2 - 3x +1) \)
Example 3: Combine End Behavior with Zeros
Sketch \( h(x) = x^4 - 2x^2 \).
Step 1: Factor: \( x^2(x^2 -2) = x^2(x - \sqrt{2})(x + \sqrt{2}) \)
Step 2: Degree = 4 (even), leading coefficient = 1 (positive) → rises on both ends
Step 3: Zeros at x = 0, ±√2 → plot intercepts
Practice Problems
- Find the end behavior of \( f(x) = -3x^5 + 2x^3 -1 \)
- Sketch the graph of \( g(x) = x^3 - 4x \)
- Determine end behavior of \( h(x) = 5x^6 - x^4 + 2 \)
- Factor and sketch \( k(x) = x^4 - x^2 \)
- Analyze end behavior and intercepts of \( m(x) = -2x^3 + x^2 - x +1 \)