Problem 40 - Algebra 2

Question

For Exercises \(39-41,\) sketch a graph of each polynomial. odd-degree polynomial function with one relative maximum and one relative minimum; the leading coefficient is negative

Step-by-Step Solution

Answer

Sketch a cubic function with a negative slope that passes through critical points at \(x = 0\) and \(x = 2\), showing one peak and one trough.

Step 1: Understand the Characteristics

We need to graph an odd-degree polynomial function that has one relative maximum and one relative minimum, with a negative leading coefficient. Odd-degree polynomial functions generally have tails going in opposite directions: one going to positive infinity and the other to negative infinity. With a negative leading coefficient, the graph will start in the positive y-direction and end in the negative y-direction.

Step 2: Choose the Simplest Example

A cubic function (degree 3), which is the simplest odd-degree polynomial, can serve as an example. For this example, we'll use the function: \[ f(x) = -x^3 + 3x^2 - 2 \] This function is constructed to have a negative leading coefficient.

Step 3: Identify Key Features

Graphically, the function \( f(x) = -x^3 + 3x^2 - 2 \) has a steep negative slope when \( x \to \infty \) and rises steeply from negative infinity as \( x \to -\infty \). The terms \(3x^2\) will create some curvature and critical points within the graph, leading to a maximum and a minimum point.

Step 4: Find Critical Points

To find the critical points, calculate the derivative \( f'(x) = -3x^2 + 6x \) and set it to zero: \(-3x^2 + 6x = 0\). Simplifying gives \(x(-3x + 6) = 0\), so the critical points are \(x = 0\) and \(x = 2\).

Step 5: Determine the Nature of Each Critical Point

Use the second derivative \(f''(x) = -6x + 6\) to find the nature of the critical points. Evaluate at \(x = 0\): \(f''(0) = 6\), indicating a local minimum. Evaluate at \(x = 2\): \(f''(2) = -6\), indicating a local maximum.

Step 6: Sketch the Graph

Start by plotting the critical points (local minimum at \(x = 0\) and local maximum at \(x = 2\)). Since the graph is that of a cubic function with a negative leading coefficient, it will begin from the top left (positive y-direction going to the left) and end in the bottom right (negative y-direction going to the right). Connect these points smoothly, ensuring it passes through the local maximum and minimum.

Key Concepts

Odd-Degree Polynomial

When dealing with an **odd-degree polynomial**, one might wonder what exactly defines it and how it behaves. In simple terms, an odd-degree polynomial is characterized by the highest exponent on its variable being an odd number. For instance, a cubic function like

• \( f(x) = -x^3 + 3x^2 - 2 \)

is of degree three, which is odd.
One of the unique features of odd-degree polynomials is how their end behaviors differ. Specifically, the graph of such a polynomial will have tails that extend in opposite directions. For example, as you move towards positive infinity on the x-axis, the graph may head towards negative infinity on the y-axis, and vice versa.
This opposing behavior contrasts with even-degree polynomials, which typically have both ends heading in the same direction as they reach infinity.

Critical Points

Discovering **critical points** within a polynomial graph is crucial for understanding its local behavior. Critical points occur where the derivative of the polynomial equals zero or is undefined.
To find critical points of a polynomial function like

• \( f(x) = -x^3 + 3x^2 - 2 \)

you first derive it: \( f'(x) = -3x^2 + 6x \).
Setting this derivative to zero shows points where the slope of the tangent to the curve is flat. In our case:

• \(-3x^2 + 6x = 0\)
• \( x = 0 \) and \( x = 2 \)

These x-values are the critical points, and they indicate where potential maximums or minimums can occur on the graph.

Leading Coefficient

The **leading coefficient** in a polynomial greatly influences its graph's overall direction. It is the coefficient in front of the term with the highest degree. For instance, in our polynomial

• \( f(x) = -x^3 + 3x^2 - 2 \)

the leading coefficient is

• -1

because of the

• \(-x^3\)

term.
A negative leading coefficient tells us that the graph starts in the positive y-direction on the left and ends in the negative y-direction on the right. This is key in predicting the overall shape and flow of the polynomial graph.

Relative Maximum

The **relative maximum** of a polynomial occurs where a temporary peak appears within a certain interval on the graph. For instance, in our polynomial function example, we determine this point using derivatives.
After setting the first derivative \( f'(x) = -3x^2 + 6x \) to zero and solving, we found critical points at \( x = 0 \) and \( x = 2 \). To identify whether they are maxima or minima, we evaluate with the second derivative:

• \( f''(x) = -6x + 6 \)
• At \( x = 2 \), \( f''(2) = -6 \), indicating a relative maximum.

At this point, the graph reaches a peak before reversing its direction.

Relative Minimum

A **relative minimum** is another crucial concept, representing a low point in the graph within a specific range. In our working example, after calculating the first derivative and finding the critical points at \( x = 0 \) and \( x = 2 \), we use the second derivative to determine their nature.
Evaluating the second derivative gives us:

• \( f''(x) = -6x + 6 \)
• At \( x = 0 \), \( f''(0) = 6 \), indicating a relative minimum.

This means the graph dips downwards before increasing again, akin to a small valley within the polynomial’s path.