Problem 54 - Algebra and Trigonometry

Question

In Exercises 41–64, a. Use the Leading Coefficient Test to determine the graph’s end behavior. b. Find the x-intercepts. State whether the graph crosses the x-axis, or touches the x-axis and turns around, at each intercept. c. Find the y-intercept. d. Determine whether the graph has y-axis symmetry, origin symmetry, or neither. e. If necessary, find a few additional points and graph the function. Use the maximum number of turning points to check whether it is drawn correctly. $$f(x)=\frac{1}{2}-\frac{1}{2} x^{4}$$

Step-by-Step Solution

Answer

End behaviour: rises to the left and right. X-intercepts are at \(x = 1\) and \(x = -1\). The y-intercept is at \(y = 1/2\). The function is neither symmetric about the y-axis nor origin. The graph of function \( f(x)=\frac{1}{2}-\frac{1}{2} x^{4} \) would be drawn by connecting additional points.

Step 1: Leading Coefficient Test

In the function \( f(x)=\frac{1}{2}-\frac{1}{2} x^{4} \), the highest degree term is \( -\frac{1}{2}x^{4} \). Since the degree is even and the leading coefficient is negative, by using the Leading Coefficient Test, we can determine that the function rises to the left and rises to the right.

Step 2: Finding the x-intercepts

To find the x-intercepts, we must set \( f(x) = 0 \). This gives us the equation \(0=\frac{1}{2}-\frac{1}{2} x^{4}\), which simplifies to \(x^{4} = 1\). Taking the fourth root gives two results \(x = 1\) and \(x = -1\). Both root are real, so the graph crosses the x-axis at these intercepts.

Step 3: Finding the y-intercept

To find the y-intercept, we substitute \( x = 0 \) into \( f(x)=\frac{1}{2}-\frac{1}{2} x^{4} \). This gives us \( f(0) =\frac{1}{2}-\frac{1}{2} \times 0^{4} =\frac{1}{2}\). Therefore, the graph crosses the y-axis at (0, 1/2).

Step 4: Determining Symmetry

Assessing if the function is symmetric about the y-axis, origin or neither, requires us to check the function for even, odd or neither. Even functions have y-axis symmetry and odd functions have origin symmetry. This function is neither even nor odd, so it has no symmetry about the y-axis or origin.

Step 5: Additional Points and Graphing

To get a better picture of the graph, choose additional points in the function to plot. Using maximum turning points, we plot these points and draw the curve of \( f(x)=\frac{1}{2}-\frac{1}{2} x^{4} \). After connecting these points smoothly, the graph of the function should appear.

Key Concepts

End Behavior

Understanding the end behavior of a polynomial function helps us predict how the graph behaves as it moves towards positive or negative infinity.
For the function \( f(x)=\frac{1}{2}-\frac{1}{2} x^{4} \), the term with the highest degree is \(-\frac{1}{2}x^{4}\). This term dominates the behavior of the graph at both extremes.

The degree of this polynomial is even, and the leading coefficient is negative. The Leading Coefficient Test tells us that:

• As \(x \to \infty\), \(f(x) \to -\infty\). This means as you move to the right on the graph, it will go down.
• As \(x \to -\infty\), \(f(x) \to -\infty\). This leads to the same downward movement as you go to the left.

Visualizing the end behavior provides a mental sketch of how the function will look like outside the center region.

X-intercepts

The x-intercepts of a polynomial function are the points where the graph intersects the x-axis.
For these points, the value of the function, \(f(x)\), is zero. To find them, solve the equation \( \frac{1}{2} - \frac{1}{2}x^{4} = 0 \).

By simplifying, you get \(x^{4} = 1\). Taking the fourth root tells us the function intersects the x-axis at:

• \(x = 1\)
• \(x = -1\)

At these points, the graph crosses the x-axis. Observing x-intercepts helps in understanding where the function changes signs and how it behaves around these intersections.

Y-intercepts

Y-intercepts occur where the graph crosses the y-axis. This is found by evaluating the function at \(x = 0\).
Plug into the equation \(f(0) = \frac{1}{2} - \frac{1}{2} \times 0^{4}\), which simplifies to \(f(0) = \frac{1}{2}\).

This means the graph crosses the y-axis at the point (0, 1/2).
Y-intercepts provide an initial starting point for graphing, establishing a connection between the function and the y-axis.
Knowing this can help in sketching graphs, often simplifying interactions with scale and other points.

Graph Symmetry

Symmetry in graphs can make sketching or understanding them easier. When a function is symmetric, it exhibits balanced traits either across the y-axis, through the origin, or has no symmetry.
To determine symmetry:

1. **Y-axis Symmetry:** - Check if \(f(x) = f(-x)\). This property indicates the graph is mirrored along the y-axis.2. **Origin Symmetry:** - Check if \(f(-x) = -f(x)\). This means the graph can be rotated 180 degrees about the origin and look the same.After assessing \(f(x) = \frac{1}{2} - \frac{1}{2}x^{4}\), neither property holds, making the function neither even nor odd. Thus, it has no particular symmetry.
Understanding symmetry helps track how a function can repeat or reflect in a predictable way. However, with this function, there isn't such symmetry to rely on.