How to Sketch the Curve of a Polynomial Function

Let’s consider the polynomial function y x^3 - 3x^2 - 2x and explore the process of sketching its curve step by step. This guide will provide a detailed explanation, including how to find roots, determine end behavior, identify critical points, evaluate the function, and conclude with a sketch of the curve.

Step 1: Find the Roots

The roots of the polynomial y x^3 - 3x^2 - 2x are the values of x where the curve intersects the x-axis. To find these, set y 0.

x^3 - 3x^2 - 2x 0

Factoring the expression, we get:

x(x^2 - 3x - 2) 0

This gives us the roots:

x 0 x -1 x -2

Step 2: Determine the End Behavior

The end behavior of a polynomial function is determined by the leading term. Since the leading term of y x^3 - 3x^2 - 2x is x^3, and the leading coefficient is positive, the end behavior is:

As x approaches positive infinity, y approaches positive infinity. As x approaches negative infinity, y approaches negative infinity.

Step 3: Find the Critical Points

Critical points help us find local maxima and minima. To find these, we first compute the first derivative of the function:

y frac{d}{dx}(x^3 - 3x^2 - 2x)

Using the power rule, we get:

y' 3x^2 - 6x - 2

Set the first derivative equal to zero and solve:

3x^2 - 6x - 2 0

Using the quadratic formula:

x frac{-b pm sqrt{b^2 - 4ac}}{2a}

Here, a 3, b -6, and c -2. Plugging in these values:

x frac{6 pm sqrt{(-6)^2 - 4(3)(-2)}}{2(3)}

x frac{6 pm sqrt{36 24}}{6}

x frac{6 pm sqrt{60}}{6} approx frac{6 pm 7.75}{6}

This gives us two solutions:

x_1 approx -0.29 x_2 approx 2.29

Step 4: Evaluate the Function at the Critical Points

Evaluate the original function at the critical points x_1 approx -0.29 and x_2 approx 2.29 to find the corresponding y-values:

When x -0.29: y approx -1.29 When x 2.29: y approx 0.29

Step 5: Determine Intervals of Increase and Decrease

Use the first derivative test to determine where the function is increasing and decreasing by checking the sign of the first derivative in the intervals determined by the roots and critical points:

Between x Function is decreasing Between x -1 and x approx -0.29: Function is increasing Between x approx -0.29 and x 0: Function is decreasing Between x 0 and x approx 2.29: Function is decreasing Between x approx 2.29 and x 3: Function is increasing

Step 6: Sketch the Curve

Plot the x-intercepts: (-2, 0), (-1, 0), and (0, 0).

Mark the y-values at the critical points: (-0.29, -1.29) and (2.29, 0.29).

Note the end behavior: the curve approaches positive infinity as x approaches positive infinity and negative infinity as x approaches negative infinity.

Draw the curve passing through the intercepts and local maxima and minima. The curve should be decreasing between x

Final Sketch

Your curve should look like a cubic polynomial with x-intercepts at -2, -1, and 0, and it will rise to the right and fall to the left.