Problem 10 - Basic Technical Mathematics with Calculus

Question

Display the graphs of the given functions on a graphing calculator. Use appropriate window settings. $$y=x^{4}-6 x^{2}$$

Step-by-Step Solution

Answer

Graph \( y = x^4 - 6x^2 \) using a window with \( x \) from \(-3\) to \(3\) and \( y \) from \(-10\) to \(10\) to cover critical points and end behavior.

Step 1: Understanding the Function

The given function is \( y = x^{4} - 6x^{2} \). It is a polynomial function of degree 4. This means it will have end behavior similar to \( y = x^4 \), with tails going toward infinity as \( x \) becomes large positive or large negative. The function will also likely have turning points, as indicated by its lower degree terms.

Step 2: Finding Critical Points

To find critical points, we need to take the derivative of the function and set it to zero. The derivative is \( \frac{dy}{dx} = 4x^3 - 12x \). Factoring gives \( 4x(x^2 - 3) = 0 \). This equation has solutions at \( x = 0 \), and \( x = \pm \sqrt{3} \). These represent the critical points where the function's rate of change is zero.

Step 3: Evaluating Function at Critical Points

Evaluate the original function \( y = x^4 - 6x^2 \) at the critical points. For \( x = 0 \), \( y = 0^4 - 6 \times 0^2 = 0 \). For \( x = \pm \sqrt{3} \), \( y = (\pm \sqrt{3})^4 - 6(\pm \sqrt{3})^2 = 9 - 18 = -9 \). So the critical points are \((0,0)\) and \((\pm \sqrt{3}, -9)\).

Step 4: Graphing the Function

Using a graphing calculator, plot the function \( y = x^4 - 6x^2 \). Set an appropriate window to fully observe the behavior around critical points and end behavior. A suitable window might be \( x \) from \(-3\) to \(3\) and \( y \) from \(-10\) to \(10\). This frame should include all the critical points and provide a clear view of the function's end behavior.

Step 5: Observing Graph Characteristics

On the graph, observe that the function passes through the origin and has local minima at \( x = \pm \sqrt{3} \) where \( y = -9 \). The graph goes upwards toward infinity on both ends, consistent with a degree 4 polynomial's end behavior. Verify that the chosen window settings display these features clearly.

Key Concepts

Polynomial Functions

Polynomial functions are equations that involve powers of a variable. They can be expressed in the standard form:

• For example, a polynomial function of degree 4 can be written as: \( y = ax^4 + bx^3 + cx^2 + dx + e \) where coefficients \( a, b, c, d, \) and \( e \) are constants.
• The degree of a polynomial is determined by the highest exponent of the variable.
• The end behavior of a polynomial function is influenced by its leading term. For a degree 4 polynomial like the one in the exercise, the ends of the graph go upwards towards infinity, similar to \( y = x^4 \).
• Polynomial functions can have several turning points and intersect the x-axis at multiple points depending on their degree.

Understand that with higher degree polynomials, the complexity of the graph increases, allowing for more intriguing behaviors such as multiple peaks and troughs.

Critical Points

Finding critical points is crucial for understanding the behavior of polynomial functions. These are the points where:

• The derivative of the function equals zero or is undefined.
• Representing locations where the function's graph can change direction.

To find critical points:

• First, calculate the derivative of the function: \( \frac{dy}{dx} \). For the function \( y = x^4 - 6x^2 \), the derivative is \( 4x^3 - 12x \).
• Set the derivative equal to zero: \( 4x(x^2 - 3) = 0 \). Solve for \( x \) to find \( x = 0 \) and \( x = \pm \sqrt{3} \).
• Critical points are where the graph feels a change, such as a peak, trough, or plateau.

By evaluating the original function at these values, you can identify the actual points on the graph, such as \((0,0)\) and \((\pm \sqrt{3}, -9)\).

Graphing Calculator

Using a graphing calculator helps in visualizing polynomial functions efficiently. These calculators allow you:

• To enter the function equation directly, such as \( y = x^4 - 6x^2 \).
• To choose appropriate window settings, which are crucial for clear visual representation. For this function, a window covering \( x \) from \(-3\) to \(3\) and \( y \) from \(-10\) to \(10\) works well.
• To zoom in or change the window for a more detailed analysis around the critical or end points.
• To trace along the graph to understand specific values and behaviors at different x-coordinates.

By plotting the function, you can see how it smoothly transitions through its turning points and asymptotes, providing a tangible reference for your calculations.

Derivative and Function Analysis

Analyzing derivatives provides insights into the behavior of polynomial functions. The derivative expresses:

• The slope or rate of change of the function at any point.
• Whether the function is increasing or decreasing at different intervals.

For our function \( y = x^4 - 6x^2 \), the derivative \( \frac{dy}{dx} = 4x^3 - 12x \) highlights slope changes:

• Where the sign of the slope changes from positive to negative or vice versa indicates potential peaks or valleys.
• Setting the derivative to zero outlines potential critical points where these changes occur.
• Examining the second derivative, \( \frac{d^2y}{dx^2} \), can help determine the nature of these points (i.e., whether they are concave up or down).

Through derivative and function analysis, you can predict and confirm the behavior of the polynomial, especially around the critical points, helping to ensure accurate graphing and deeper understanding of the function.