Question

Sketch a graph of the function f(x)=x³+x²-4x-4 by finding the zeros and determining the sign of the function between zeros. Explain how the structure of the equation helps guide your sketch.

Answer 1

Answer and Step-by-step explanation:

f(x) = x³ + x² - 4x - 4

Using the rational roots theorem we can determine that 2 is a root of this equation:

f(2) = 2³ + 2² - 4.2 - 4 = 8 + 4 - 8 - 4 = 0

By using Ruffini's rule we can find the others roots

2 | 1 1 -4 -4

1 3 2 0

x² + 3x + 2 = 0

S = -3 P = 2

x' = -1

x" = -2

roots are: -2 -1 2

sign before -2: f(-3) = -10 < 0

sign between -2 and -1: f(-1.5) = 0.875 > 0

sign between -1 and 2: f(0) = -4 < 0

sign after 2: f(3) = 20 > 0

This way, we can see the behaviour of the function