Problem 26 - 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. $$f(x)=x^{4} ;(0, \infty)$$

Step-by-Step Solution

Answer

The function is increasing on the interval \((0, \infty)\).

Step 1: Understand the Function

The given function is \( f(x) = x^4 \). This is a polynomial function with the highest degree term of \( x^4 \). Such functions generally follow the shape of a parabola as \( x \) approaches infinity.

Step 2: Analyze the Behavior of the Function

Since the interval is \( (0, \infty) \), we're focusing on the behavior of the function to the right of the y-axis, starting from just above zero. In this region, the higher power of \( x \) in \( x^4 \) will dominate, implying the function should closely align with a parabola opening upwards.

Step 3: Graph the Function

Use a graphing calculator to plot \( f(x) = x^4 \). Set your viewing window to capture the interval from \( x = 0 \) onwards. Observe the graph's trajectory to determine how it behaves as \( x \) increases.

Step 4: Determine the Behavior from the Graph

As you trace from left to right starting near zero, observe that the graph of \( x^4 \) goes upwards. This consistent upward movement indicates that the function is increasing on the interval \( (0, \infty) \).

Key Concepts

Graphing Calculators

Graphing calculators are powerful tools that help visualize the behavior of mathematical functions, such as polynomial functions like \( f(x) = x^4 \). By graphing a function, students can easily see how the function behaves over specific intervals. Set the viewing window of your graphing calculator to focus on the desired interval—in this case, \( x \) from 0 to some large positive number. This helps to better understand how the function acts in that segment. Using the trace function, you can move along the graph of the function, observing its formation and noting any changes in direction or shape. Visualizing the graph in this way makes it easier to determine if the function increases or decreases over the interval.

Function Behavior

Understanding the behavior of a function involves recognizing how the output values of the function change as the input values vary. For polynomial functions like \( f(x) = x^4 \), which have even exponents, the graph generally forms a U-shape opening upwards. This is because the term with the highest exponent, \( x^4 \), dominates the behavior as \( x \) becomes large, especially on positive intervals like \( (0, \, \infty) \).

• As \( x \) increases, the value of \( f(x) \) becomes larger, displaying an upward trend.
• For different types of polynomials—those with odd exponents or negative coefficients—the behavior might significantly differ.

Recognizing these trends enables you to anticipate the nature of other polynomial functions simply by examining their highest degree terms.

Increasing and Decreasing Functions

Determining whether a function is increasing or decreasing is crucial for understanding the overall trend of the function. In simple terms:

• A function is increasing if the value of \( f(x) \) becomes larger as \( x \) increases.
• Conversely, a function is decreasing if \( f(x) \) gets smaller as \( x \) goes up.

In the interval \( (0, \infty) \), examining the graph of \( f(x) = x^4 \) through a graphing calculator reveals that it rises upward, indicating it is increasing throughout that entire stretch. Recognizing an increasing or decreasing nature is essential because it provides valuable insights into function characteristics, like potential local minima or maxima, which are points where the direction changes.