Problem 25 - A Graphical Approach to Precalculus with Limits

Question

Graph each finction in the standand viewing window of your calculator, and trace from left to right along a representative portion of the specified interval. Then fill in the blank of the following sentence with either increasing or decreasing. $$f(x)=x^{4} ;(-\infty, 0)$$

Step-by-Step Solution

Answer

The function is decreasing on \((-\infty, 0)\).

Step 1: Understanding the function

The function given is \( f(x) = x^4 \). This is a polynomial function with an even degree. When the degree is even, the shape of the graph is symmetrical, and generally for large values of \( x \) in either direction, the function approaches positive infinity.

Step 2: Analyzing the interval

The specified interval is \( (-\infty, 0) \). This interval includes all real numbers less than zero. We are interested in understanding the behavior of the function as \( x \) moves from negative infinity towards zero.

Step 3: Graphing the function

To graph the function \( f(x) = x^4 \) in your calculator, set your viewing window to include negative values of \( x \) up to some positive values to ensure you cover the interval \((-\infty, 0)\). A typical standand viewing window can range from \(-10\) to \(10\) for both \( x \) and \( y \) axes.

Step 4: Tracing the function

Use the trace feature on your calculator. Start tracing from the left, which is at very large negative \( x \) values, and move towards zero. Observe how the values of \( f(x) \) change as \( x \) increases in this interval.

Step 5: Determining the behavior

As \( x \) increases from negative values approaching zero, the graph of \( y = x^4 \) decreases. This means that \( f(x) = x^4 \) is decreasing on the interval \((-\infty, 0)\).

Key Concepts

Graphing Calculators

Graphing calculators are amazing tools that help visualize mathematical functions by plotting their graphs. They make it easier to understand the function’s behavior just by looking at the shape of the graph. To graph a function like \( f(x) = x^4 \), start by setting up your calculator's viewing window. This ensures that you have a good view of the portion of the graph you’re interested in.

• For this particular exercise, you want the graph to cover the negative part of the x-axis, up to zero.
• A typical window setting could be from \(-10\) to \(10\) for both x and y axes.

Once the window is set, simply input the function and press the graph button. The calculator will display the curve of \( f(x) = x^4 \), and from this, you can use the trace feature. This feature lets you move along the graph, making it simple to see how the function values change as \( x \) values increase.

Function Behavior

Understanding a function's behavior involves observing how its values change as \( x \) changes. For the polynomial function \( f(x) = x^4 \), the behavior is unique because it has a degree of 4, which is even.Between any two points on a function, you can determine if the function is increasing or decreasing:

• Increasing: As \( x \) goes up, \( f(x) \) also goes up.
• Decreasing: As \( x \) goes up, \( f(x) \) goes down.

For \( f(x) = x^4 \), focusing on the interval \((-\infty, 0)\), you notice that as \( x \) moves from a larger negative value towards zero, the values of \( f(x) \) actually decrease. This means that \( f(x) \,=\, x^4 \) is decreasing over this interval. The trace feature on your graphing calculator makes it easy to see this pattern.

Polynomial Degree

The degree of a polynomial is vital in determining the graph's overall shape and how the function behaves at the ends of the x-axis.For \( f(x) = x^4 \):

• The degree is 4, which is even.
• Even degree polynomial graphs are symmetrical about the y-axis.

This means that both ends of the graph generally point in the same direction. For even powers like 4, as \( |x| \) becomes very large, both ends head towards infinity. In our case, while the function decreases in the interval \((-\infty, 0)\), each end of the graph ultimately heads upwards.