Describe & Sketch the graph: f(x) = x^3 + 2x^2 - 4x - 8

Answer

To describe and sketch the graph of f(x) = x^3 + 2x^2 - 4x - 8, we follow several steps: 1. End Behavior: The function is a cubic polynomial (degree 3) with a positive leading coefficient (+1). This means as x approaches infinity, f(x) approaches infinity, and as x approaches negative infinity, f(x) approaches negative infinity. The graph goes from the bottom-left to the top-right. 2. Y-intercept: Set x = 0. f(0) = 0^3 + 2(0)^2 - 4(0) - 8 = -8. The y-intercept is (0, -8). 3. X-intercepts (Roots): Factor the polynomial by grouping: f(x) = (x^3 + 2x^2) - (4x + 8) = x^2(x + 2) - 4(x + 2) = (x^2 - 4)(x + 2) = (x - 2)(x + 2)(x + 2) = (x - 2)(x + 2)^2. Setting f(x) = 0 gives x = 2 and x = -2. 4. Multiplicity: The root at x = 2 has a multiplicity of 1, so the graph crosses the x-axis there. The root at x = -2 has a multiplicity of 2, so the graph touches the x-axis and turns back (it is tangent to the axis). 5. Sketching: Start in the third quadrant, come up to touch the x-axis at x = -2, dip down to pass through (0, -8), and then cross the x-axis at x = 2 heading into the first quadrant.