Problem 83 - A Graphical Approach to Precalculus with Limits

Question

Use the shading capabilities of a graphing calculator to graph each inequality or system of inequalities. $$\begin{aligned}&y \geq 2^{x}\\\&y \leq 8\end{aligned}$$

Step-by-Step Solution

Answer

The solution is the shaded area where \( y \geq 2^x \) and \( y \leq 8 \).

Step 1: Rewrite each inequality

Look at the given inequalities: \( y \geq 2^x \) and \( y \leq 8 \). These describe relationships between \( y \) and \( x \) that we'll plot on a graph. They've already been expressed in forms suitable for graphing.

Step 2: Graph the function \( y = 2^x \)

On the graphing calculator, plot the equation \( y = 2^x \). This is an exponential function that starts at \( y = 1 \) when \( x = 0 \) and increases as \( x \) increases.

Step 3: Identify the region for \( y \geq 2^x \)

The inequality \( y \geq 2^x \) means we want the region above the curve \( y = 2^x \). On your graphing calculator, use the shading feature to shade this area.

Step 4: Graph the horizontal line \( y = 8 \)

Plot the line \( y = 8 \) across the graph. This is a horizontal line where \( y \) is constant at 8, no matter the value of \( x \).

Step 5: Identify the region for \( y \leq 8 \)

The inequality \( y \leq 8 \) refers to the area below or on the line \( y = 8 \). Use the calculator's shading function again to shade below this horizontal line.

Step 6: Find the intersection area

The solution to the system of inequalities is where the shading overlaps. Identify the region that is both above \( y = 2^x \) and below \( y = 8 \). This will be the shaded area between these two boundary lines.

Key Concepts

Exponential Functions

Exponential functions involve a constant base raised to a variable exponent. In the context of the given exercise, the function is represented as \( y = 2^x \). This means the value of \( y \) changes exponentially with respect to \( x \). Here's how the function behaves:

• For \( x = 0 \), \( y = 2^0 = 1 \).
• As \( x \) increases, \( y \) increases rapidly since it follows a geometric progression.
• Conversely, as \( x \) decreases, the value of \( y \) approaches zero but never becomes negative or zero, since the base (2) is positive.

Exponential functions like these are crucial in many real-world applications, such as calculating compound interest, population growth, or exponential decay. When graphing, it's essential to understand how the function's rapid increase affects the shading in inequality problems.

Graphing Calculator Usage

A graphing calculator is a powerful tool for visualizing mathematical functions and inequalities. When tackling inequalities like \( y \geq 2^x \) and \( y \leq 8 \), a graphing calculator can help plot these accurately.To use your graphing calculator effectively:

• Start by inputting the inequality equations as functions: \( y = 2^x \) and \( y = 8 \).
• Use the graphing feature to plot these equations. For \( y = 2^x \), expect an exponential curve. For \( y = 8 \), expect a straight horizontal line.
• Utilize the shading tool to highlight the areas that satisfy the inequalities. This is done by shading above the curve for \( y \geq 2^x \) and below the line for \( y \leq 8 \).

This visual aid helps quickly identify the intersection of shaded regions, which are critical for finding solutions to complex inequality systems.

Intersection of Regions

In the context of graphing inequalities, finding the intersection of regions involves identifying where two or more shaded areas overlap. This is crucial for determining the solution to a system of inequalities.Here, we look at the intersection of the areas defined by \( y \geq 2^x \) and \( y \leq 8 \):

• The first region is above the exponential curve \( y = 2^x \), representing all points where \( y \) is greater than or equal to \( 2^x \).
• The second region is below the horizontal line \( y = 8 \), including all points where \( y \) is less than or equal to 8.
• The intersection of these regions lies between the curve and the line, where both inequalities are true at the same time.

Understanding the intersection shows the common solution area that satisfies both inequalities, providing a visual and logical comprehension of what the system of inequalities represents. By identifying this overlap on a graph, you can see which values meet all given conditions.