Problem 108 - College Algebra Essentials

Question

use a graphing utility to graph each function. Use \(a[-5,5,1]\) by \([-5,5,1]\) viewing rectangle. Then find the intervals on which the function is increasing, decreasing, or constant. $$ g(x)=\left|4-x^{2}\right| $$

Step-by-Step Solution

Answer

Upon graphing the function \( g(x)= \left|4-x^{2}\right| \), it has been observed that the function is increasing in the interval from \(-\infty\) to \(0\). At \(0\), it reaches its maximum value of \(4\) and then starts decreasing in the interval from \(0\) to \(\infty\).

Step 1: Plotting the Graph

Utilize a graphing utility to plot the function. The function \(g(x)\) can be rewritten as \( y = \left|4-x^{2}\right|\) and both \(x\) and \(y\) in the viewing rectangle range from \(-5\) to \(5\) with a step size of \(1\). This gives a clear visualization of the function within the given interval.

Step 2: Observing the Graph

After the graph is plotted, careful observation is necessary. The function graph will appear as a mirror image with respect to y-axis (symmetric along y-axis), and the vertex of parabola at \(x=0\).

Step 3: Identifying intervals

From \(-\infty\) to \(0\), the function \(y\) is increasing; At \(0\), the function reaches its maximum value \(4\); From \(0\) to \(\infty\), the function \(y\) is decreasing.

Key Concepts

Graphing Utilities

Graphing utilities are powerful tools that aid in visualizing mathematical functions. Using software or calculators designed for graphing can help students quickly plot complex functions like absolute value functions. For this exercise, we use a viewing window from

• to 5 for both the x and y axes, with an increment of 1.
• The window size helps to ensure that all critical features of the function are visible.
• Graphing utilities often allow zooming in and out, as well as shifting the graph perspective.

These tools enable students to better grasp how changes in a function's formula influence its graph. By visually assessing the function, students can identify characteristics like symmetry and intercepts easily, saving significant time compared to manual plotting.

Function Intervals

Understanding function intervals involves determining where a function increases, decreases, or stays constant. When analyzing the function \( g(x) = \left| 4 - x^2 \right| \), we can split the analysis around critical points.

• From \( -\infty \) to \( 0 \), the function increases.
• At \( x = 0 \), the function reaches its peak value.
• Then, it starts decreasing from \( 0 \) to \( \infty \).

The behavior of functions across intervals tells us about their overall direction. Observing interval behavior assists in making conjectures about the features of a function graph. Understanding these patterns is crucial, as it lays the groundwork for more advanced topics like calculus.

Absolute Value Functions

Absolute value functions, such as \( g(x) = \left| 4 - x^2 \right| \), display unique characteristics due to the absolute value operator. This operator causes the graph to take only non-negative values, even when the input might typically produce negative results. Key aspects include:

• Symmetry about the y-axis, making the graph a mirror image.
• The vertex, where the function changes from increasing to decreasing, sits at the peak.
• Effectively reflects any portion of the graph below the x-axis to be above it.

These properties contribute to the distinctive V-shape of basic absolute value graphs, though transformations like shifts and stretches can alter appearance but not fundamental behavior.

Parabolas

Parabolas are U-shaped graphs that are characteristic of quadratic functions, appearing both in absolute value functions and standard quadratic equations. Despite being wrapped inside an absolute value, the equation \( y = 4 - x^2 \) is a classic downward-opening parabola.

• When you add the absolute value operator, the bottom part of the parabola flips upwards at the x-axis, forming a bowl shape.
• The vertex of this parabola is at the point \( (0, 4) \), which also serves as the highest point of the absolute value graph.
• The arms of the parabola widen as they extend along the x-axis.

Understanding the structure of parabolas helps in comprehending how changes to the standard formula (like coefficients or absolute operators) affect the overall graph appearance. This understanding is essential for various applications, including physics and engineering.