Question

Graph each polynomial function. $$ f(x)=2 x^{3}-x^{2}+2 x-1 $$

Short answer

Identify coefficients and determine polynomial behavior. Use derivatives for critical and inflection points. Plot key points to sketch the graph.

Step-by-step solution

- Identify the Polynomial

The given polynomial is ewline ewline f(x) = 2x^3 - x^2 + 2x - 1. Identify the coefficients which are: a=2, b=-1, c=2, and d=-1.

- Determine the End Behavior

Given that the highest degree is 3 (cubic function), and the leading coefficient is positive (2), the end behavior of the graph will be: - As x approaches ∞, f(x) approaches ∞- As x approaches -∞, f(x) approaches -∞.

- Find the Derivatives

Find the first derivative f'(x) = 6x^2 - 2x + 2 to locate critical points. Find the second derivative f''(x) = 12x - 2 to determine concavity and inflection points.

- Solve for Critical Points

Set the first derivative equal to zero to find critical points: 6x^2 - 2x + 2 = 0. Solve this quadratic equation for x.

- Solve for Inflection Points

Set the second derivative equal to zero to find inflection points: 12x - 2 = 0 Solve this for x.

- Use Test Points to Determine the Shapes

Test values around the critical points and inflection points to determine the intervals where the function is increasing or decreasing, as well as concave up or down.

- Plot Key Points and Sketch the Graph

After determining the critical and inflection points, plot these points. Next, using the behavior in those intervals, sketch the overall shape of the polynomial on the graph.

Key concepts

End Behavior

When graphing polynomial functions, understanding the end behavior is crucial. It tells you how the function behaves as the input values (x) become very large or very small.
For the polynomial function given, \( f(x) = 2x^3 - x^2 + 2x - 1 \), the highest degree term is \( 2x^3 \). This determines the end behavior.
The degree is 3 (odd) and the leading coefficient is positive (2).

• As \( x \) approaches ∞, \( f(x) \) also approaches ∞.
• As \( x \) approaches -∞, \( f(x) \) approaches -∞.

This means the graph starts in the third quadrant (down and to the left) and ends in the first quadrant (up and to the right). Always check the leading term and its coefficient to get a sense of the general direction of the graph.

Derivatives

Derivatives help us understand the rate of change of the function. They are used to find critical points and inflection points.
The first derivative of the given polynomial \( f(x) \) is \( f'(x) = 6x^2 - 2x + 2 \). This helps in identifying where the function is increasing or decreasing.
The second derivative \( f''(x) = 12x - 2 \) is used to find concavity and inflection points.

• The first derivative, \( f'(x) \), is set to zero to find critical points.
• The second derivative, \( f''(x) \), is set to zero to find inflection points.

Understanding how the derivatives work gives deeper insights into the shape and behavior of the graph.

Critical Points

Critical points occur where the first derivative equals zero or is undefined. These are points where the function changes direction (i.e., from increasing to decreasing or vice versa).
For the polynomial \( f(x) = 2x^3 - x^2 + 2x - 1 \), we set the first derivative \( f'(x) = 6x^2 - 2x + 2 \) equal to zero:
\( 6x^2 - 2x + 2 = 0 \). Solve this quadratic equation to find the critical points.
These critical points help to identify the local maxima and minima of the function. Test values around the critical points using the first derivative to determine if the function is increasing or decreasing in those intervals.
This analysis is vital in sketching a more accurate graph.

Inflection Points

Inflection points mark where the function changes its concavity (from concave up to concave down or vice versa). To find these, we use the second derivative.
For the polynomial function given, we set the second derivative \( f''(x) = 12x - 2 \) equal to zero:
\( 12x - 2 = 0 \). Solve for \( x \) to get the inflection point.
At these inflection points, the concavity of the function changes:

• If \( f''(x) > 0 \), the function is concave up.
• If \( f''(x) < 0 \), the function is concave down.

Understanding inflection points helps in sketching the curve accurately and knowing where the graph bends.
These points make the graphing of polynomial functions more precise and informative.