Problem 25 - A Graphical Approach to College Algebra

Question

Graph each function in the standard 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. Over the interval specified, this function is __________. $$f(x)=x^{4} ;(-\infty, 0)$$

Step-by-Step Solution

Answer

The function is decreasing over the interval \((-\infty, 0)\).

Step 1: Understand the Function

The function given is \(f(x) = x^4\), which is a polynomial function. Its graph is symmetric about the y-axis because it only contains even powers of \(x\).

Step 2: Analyze the Interval

The specified interval is \((-\infty, 0)\), which means we are interested in the behavior of the function for negative values of \(x\).

Step 3: Graph the Function

Using your calculator, graph \(f(x) = x^4\). You will notice that the curve is a parabola-like shape but steeper, opening upwards, because it is a quartic (fourth-degree) function.

Step 4: Trace on the Graph

Trace the graph from left to right within the interval \((-\infty, 0)\). As you trace from very large negative \(x\) values towards \(x = 0\), observe how the function behaves.

Step 5: Determine the Function's Behavior

Notice that as \(x\) moves from \(-\infty\) to 0, the value of \(f(x)\) decreases approaching closer to zero. Therefore, the function is decreasing in this interval.

Key Concepts

Graphing Polynomial Functions

The function given in the exercise is a polynomial function, specifically a quartic function, represented by \(f(x) = x^4\). Understanding polynomial function graphing is all about seeing how they behave on a coordinate plane. Graphing this function might seem tricky, but it becomes easier when we understand its characteristics:

• Symmetry: The graph of \(x^4\) is symmetric about the y-axis because it contains only even powers of \(x\). This means the graph looks the same on both sides of the y-axis.
• Shape: A quartic function graph resembles a U-shape, akin to a parabola but wider and flatter near the origin, steepening as you move away from the center.
• Graphing: When using a graphing calculator, you will see these features prominently, with the arms moving up towards positive infinity.

Graphing such functions not only helps in visual analysis but also in verifying characteristics such as intervals of increase or decrease.

Interval Analysis in Polynomial Functions

Interval analysis involves examining how a function behaves within a specified range. For \(f(x) = x^4\), we focus on the interval \((-\infty, 0)\). This means we're looking at what's happening along the x-axis for negative values.

• Direction of the Interval: Since we are looking from \(-\infty\) up to zero, our analysis starts from large negative numbers.
• Impact on Function Values: As \(x\) becomes larger negative, \(x^4\) becomes larger positive. But as it creeps towards zero from the negative side, \(x^4\) decreases.
• Behavior over the Interval: In this interval, the function is effectively decreasing. This is observed from the graph where the curve slopes downwards as it approaches zero.

Interval analysis provides the insight needed to describe the segment of the graph and infer important behaviors like increasing or decreasing.

Understanding the Behavior of Polynomial Functions

Behavior analysis of polynomial functions tells us about their increasing or decreasing nature over specific intervals. The function \(f(x) = x^4\) is a wonderful example of such an analysis.

• Decreasing Interval: Over the interval \((-\infty, 0)\), the function decreases steadily which means the y-values (outputs of the function) get smaller as the x-values (inputs) approach 0 from negatives.
• Why It Decreases: This occurs because the even power function approaches its vertex at \(x=0\) symmetrically, so when you move from negative large to zero, the outputs reduce.
• Predictive Analysis: Understanding this behavior is crucial because it allows predictions about values of the function at any point within this interval.

Recognizing these behavioral patterns aids not just in solving textbook problems but equips students to generalize these concepts to other mathematical functions too.