Problem 53 - College Algebra and Calculus: An Applied Approach

Question

Analyzing a Graph In Exercises \(47-58\), analyze the graph of the function algebraically and use the results to sketch the graph by hand. Then use a graphing utility to confirm your sketch. $$f(x)=x^{3}-9 x$$

Step-by-Step Solution

Answer

The roots of the function are \(0\), \(3\) and \(-3\). The function is increasing for the values of \(x\) less than \(-sqrt(3)\) and greater than \(sqrt(3)\), and it is decreasing when \(x\) is between \(-sqrt(3)\) and \(sqrt(3)\). From here, a w-shaped graph can be hand-drawn and then confirmed using a graphing utility.

Step 1: Identify Roots

Set \(f(x)=0\). Solve \(x^{3}-9x=0\). By factorizing, \(x(x^2-9)=0\). Therefore, three roots can be identified: \(x=0\), \(x=3\), \(x=-3\).

Step 2: Find Extreme Points

The first derivative of the function is \(f'(x)=3x^{2}-9\). Now, set \(f'(x)=0\), then solve \(3x^{2}-9=0\). The solutions are \(x=sqrt(3)\) and \(x=-sqrt(3)\). These points are the local maxima and minima of the function.

Step 3: Determine The Function's Increasing or Decreasing Nature

For \(x > sqrt(3)\), \(f'(x)=3x^{2}-9\) is positive, therefore the function increases. For \(-sqrt(3) < x < sqrt(3)\), \(f'(x)=3x^{2}-9\) is negative, therefore the function decreases. For \(x < -sqrt(3)\), \(f'(x)=3x^{2}-9\) is positive, therefore the function increases.

Step 4: Sketching the Graph

Now with all of this information, a sketch of the graph can be drawn. The roots are at \(x=0\), \(x=3\), and \(x=-3\). The function is increasing for \(x<-sqrt(3)\) and \(x>sqrt(3)\), and decreasing for \(-sqrt(3)

Step 5: Confirming with a Graphing Utility

To confirm your sketch, draw the function \(f(x)=x^{3}-9x\) using any graphing utility and verify the behavior at key points is as expected.

Key Concepts

Roots of a Function

Understanding the roots of a function is pivotal in graph analysis. These roots, also known as the zeros of the function, are the x-values at which the function equals zero. For the function given, we have:

• Function: \(f(x) = x^3 - 9x\)
• To find the roots, set \(f(x) = 0\)

By rearranging the equation, we get \(x(x^2 - 9) = 0\). This factorization reveals the roots \(x = 0\), \(x = 3\), and \(x = -3\). These points are where the graph will intersect the x-axis, crucial for sketching the behavior of the function graph.
These roots provide not just intersections but also reveal symmetry in the function. Understanding these will help you sketch the graph with confidence.

Derivative Analysis

Derivative analysis helps determine the rate of change of a function. The first derivative, \(f'(x)\), is vital for identifying critical points like local maxima and minima. For \(f(x) = x^3 - 9x\), the derivative is computed as:

• \(f'(x) = 3x^2 - 9\)

Setting \(f'(x) = 0\), we solve \(3x^2 - 9 = 0\) and find \(x = \sqrt{3}\) and \(x = -\sqrt{3}\). These critical values help locate the peaks and troughs of the graph, indicating where the function changes direction from increasing to decreasing or vice versa.
It’s important to analyze subintervals around these points to determine where the function is increasing or decreasing, giving deeper insight into the graph’s structure.

Function Behavior

Analyzing function behavior involves determining where the function is increasing or decreasing. Using derivative analysis, we've identified that:

• For \(x > \sqrt{3}\), since \(f'(x) = 3x^2 - 9\) is positive, the function is increasing.
• For \(-\sqrt{3} < x < \sqrt{3}\), \(f'(x) = 3x^2 - 9\) is negative, so the function is decreasing.
• For \(x < -\sqrt{3}\), \(f'(x) = 3x^2 - 9\) is positive, which means the function increases again.

This change in direction forms a w-shaped curve on the graph. Recognizing these intervals helps in understanding the graph’s shape and is crucial for sketching and predicting future behavior.

Sketching Graphs

Once you know the roots and understand the function behavior, sketching becomes straightforward. Begin by plotting the roots:

• X-intercepts at \(x = -3, 0,\) and \(3\)
• Mark critical points at \(x = \pm \sqrt{3}\)

From here, use the increasing and decreasing behavior:

• The curve decreases from \(-\sqrt{3}\) to \(\sqrt{3}\).
• Increases before \(-\sqrt{3}\) and beyond \(\sqrt{3}\).

This information creates a w-shape, with the trough between \(-\sqrt{3}\) and \(\sqrt{3}\).
Finally, confirm your sketch with a graphing utility, ensuring accuracy against plotted points and behavior, allowing you to visualize the full graph dynamically.

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Key Concepts

Roots of a Function

Understanding the roots of a function is pivotal in graph analysis. These roots, also known as the zeros of the function, are the x-values at which the function equals zero. For the function given, we have:

• Function: \(f(x) = x^3 - 9x\)
• To find the roots, set \(f(x) = 0\)

By rearranging the equation, we get \(x(x^2 - 9) = 0\). This factorization reveals the roots \(x = 0\), \(x = 3\), and \(x = -3\). These points are where the graph will intersect the x-axis, crucial for sketching the behavior of the function graph.
These roots provide not just intersections but also reveal symmetry in the function. Understanding these will help you sketch the graph with confidence.

Derivative Analysis

Derivative analysis helps determine the rate of change of a function. The first derivative, \(f'(x)\), is vital for identifying critical points like local maxima and minima. For \(f(x) = x^3 - 9x\), the derivative is computed as:

• \(f'(x) = 3x^2 - 9\)

Setting \(f'(x) = 0\), we solve \(3x^2 - 9 = 0\) and find \(x = \sqrt{3}\) and \(x = -\sqrt{3}\). These critical values help locate the peaks and troughs of the graph, indicating where the function changes direction from increasing to decreasing or vice versa.
It’s important to analyze subintervals around these points to determine where the function is increasing or decreasing, giving deeper insight into the graph’s structure.

Function Behavior

Analyzing function behavior involves determining where the function is increasing or decreasing. Using derivative analysis, we've identified that:

• For \(x > \sqrt{3}\), since \(f'(x) = 3x^2 - 9\) is positive, the function is increasing.
• For \(-\sqrt{3} < x < \sqrt{3}\), \(f'(x) = 3x^2 - 9\) is negative, so the function is decreasing.
• For \(x < -\sqrt{3}\), \(f'(x) = 3x^2 - 9\) is positive, which means the function increases again.

This change in direction forms a w-shaped curve on the graph. Recognizing these intervals helps in understanding the graph’s shape and is crucial for sketching and predicting future behavior.

Sketching Graphs

Once you know the roots and understand the function behavior, sketching becomes straightforward. Begin by plotting the roots:

• X-intercepts at \(x = -3, 0,\) and \(3\)
• Mark critical points at \(x = \pm \sqrt{3}\)

From here, use the increasing and decreasing behavior:

• The curve decreases from \(-\sqrt{3}\) to \(\sqrt{3}\).
• Increases before \(-\sqrt{3}\) and beyond \(\sqrt{3}\).

This information creates a w-shape, with the trough between \(-\sqrt{3}\) and \(\sqrt{3}\).
Finally, confirm your sketch with a graphing utility, ensuring accuracy against plotted points and behavior, allowing you to visualize the full graph dynamically.