Question

Sketch the graph of the function. \(f(x)=x^{4}-4 x^{2}\)

Short answer

The graph of \(f(x)=x^{4}-4 x^{2}\) has zeros at \(x=0,\ x=\pm2\), and has a minimum point at \(x=0\) and maximum points at \(x=\pm2\). At \(x=0\), the function changes from negative to positive as \(x\) moves from negative values to positive values. At \(x=\pm2\), the function changes from positive to negative as \(x\) moves further away from 0.

Step-by-step solution

Identify the zeros of the function

To find the zeros of the function, you set the function equal to zero and solve for x. \(f(x)=x^{4}-4 x^{2} =0\). This can be further simplified by taking out a common factor \(x^2\), \(x^2 * (x^2 - 4) =0\). Solving for \(x\) gives \(x=0, x= 2, x= -2\). These are the points where the graph will intersect the x-axis.

Determine the behavior at the zeros

At \(x=0\), the function is positive for \(x<0\) and \(x>0\), and is negative for \(x < -2\), \( -2< x < 0\),\(\ 0< x < 2\), and \(x > 2\). This will determine the shape of the graph around the intersection points.

Find the maximum and minimum points

The maxima and minima of the function are found by taking the derivative of the function \(f'(x)= 4x^{3}-8 x\), and setting this equal to zero. Solving for the derivative gives the minimum value at \(x=0\) and maximum values at \(x=2, x= -2\). This provides three turning points for the graph.

Sketch the graph

With the zeros, behavior around the zeros, and maximum and minimum points, you can now sketch the rough graph of the function. The graph will intersect the x-axis at \(x=0,\ x=\pm2\) and will have turning points at \(x=0,\ x=\pm2\).

Key concepts

Finding Zeros

To graph a polynomial function like \( f(x) = x^4 - 4x^2 \), finding the zeros is your first step. Zeros occur where the graph intersects the x-axis. This means that \( f(x) = 0 \). You can simplify the equation by factoring to find these points. Here, we factor the equation as \( x^2 (x^2 - 4) = 0 \). This gives two factors, \( x^2 = 0 \) and \( x^2 - 4 = 0 \), leading you to the solutions \( x = 0, 2, -2 \). These values are the zeros of your function.

Finding zeros is crucial because these are the points where the polynomial changes sign. This indicates critical points where the graph crosses or touches the x-axis. Not every zero is a crossing point; sometimes the graph will touch and turn around back in the direction it came from. In this exercise, the zeros \( x = 0, 2, -2 \) are essential for understanding and sketching the polynomial graph.

Behavior of Functions

Understanding the behavior of a function near its zeros helps you know how the graph will look. For \( f(x) = x^4 - 4x^2 \), knowing the sign of the function in intervals divided by the zeros is essential.

For example:

• When \( x < -2 \), \( f(x) \) is positive because both \( (x^2) \) and \( (x^2 - 4) \) are positive.
• When \( -2 < x < 0 \), \( f(x) \) becomes negative because \( (x^2 - 4) \) is negative.
• When \( 0 < x < 2 \), \( f(x) \) is still negative, but switches after \( x = 2 \), where the function becomes positive again.

These variations in sign indicate rises and falls in the graph, signaling possible turns at these points. These behaviors help determine the shape and progression of your graph by identifying which sections of \( x \) correspond with increases or decreases in \( f(x) \).

A clear understanding of such behavior allows for accurate sketches and predictions of where the graph is positive or negative, creating a fuller picture of the polynomial function's dynamics.

Turning Points

Turning points in a polynomial function occur where the derivative is zero, indicating a local maximum or minimum. For \( f(x) = x^4 - 4x^2 \), you derive it to get \( f'(x) = 4x^3 - 8x \). Setting this equal to zero, \( 4x(x^2 - 2) = 0 \), results in \( x = 0, \pm2 \) being your solutions, which are your turning points.

These points are significant because they indicate where the graph stops increasing and starts decreasing, or vice versa.

• At \( x = -2 \) and \( x = 2 \), you have local maximum points.
• At \( x = 0 \), you find a local minimum point.

These can be visual cues when sketching the graph, highlighting areas of interest where changes in direction occur. Understanding turning points is crucial since they define the apexes or valleys of the graph, thereby structuring the overall flow and layout when sketching.

Remember, turning points are central to determining the largest and smallest values within specific intervals. They empower you to sketch the drastic shifts and maintain overall accuracy in graph depiction. Analyzing derivative results helps to pinpoint these critical points on your graph.