Problem 22 - Precalculus with Limits

Question

In Exercises 19-28, use a graphing utility to graph the inequality. $$y \leq 2^{2 x-0.5}-7$$

Step-by-Step Solution

Answer

The graph of the inequality \(y \leq 2^{2x-0.5}-7\) is the exponential curve \(f(x) = 2^{2x-0.5}-7\), along with the area under this curve. The shading represents the set of all \(y\) values that satisfy the given inequality.

Step 1: Write down the inequality function

The inequality given is \(y \leq 2^{2x-0.5}-7\). The function from this inequality is \(f(x) = 2^{2x-0.5}-7\).

Step 2: Graph the function

Using a graphing tool, plot the function \(f(x) = 2^{2x-0.5}-7\) on a graph. It's important to remember this is an exponential function with base 2, so the curve shoots upward for positive \(x\) and approaches the x-axis (without touching it) for negative \(x\). Also note, the '-7' shifts the entire function 7 units down the y-axis.

Step 3: Shade the area for the inequality solution

Since the inequality is \(y \leq 2^{2x-0.5}-7\), the solution consists of all the \(y\) values less than or equal to this function. Therefore, shade the area under the curve (including the curve, because it's '\(\leq\)') to represent the solution to the inequality.

Key Concepts

Understanding Exponential Functions

Exponential functions are powerful mathematical expressions wherein variables appear as exponents. The general form of an exponential function is \[ f(x) = a imes b^{x+c} - d \]where:

• \(a\) is a constant that can stretch or compress the graph vertically.
• \(b\) is the base, dictating how steeply the graph increases or decreases.
• \(x\) is the exponent, representing the variable input.
• \(c\) shifts the graph left or right.
• \(d\) moves the graph up or down.

In the exercise, the function is \(f(x) = 2^{2x-0.5}-7\). Here, the base is 2, implying exponential growth. The expression \(2x-0.5\) as an exponent affects the function’s growth rate and horizontal position. The term \(-7\) shifts the entire function 7 units downward on the y-axis. Consequently, the exponential growth curve starts below where it normally would if it wasn't shifted, altering its intercepts and overall appearance.

Using a Graphing Utility

Graphing utilities are tools, such as calculators or software programs, that assist in plotting mathematical functions effortlessly. They provide a visual representation of equations and inequalities, simplifying complex calculations and offering instant feedback on changes.In our exercise, a graphing utility helps to plot the exponential equation, \(f(x) = 2^{2x-0.5}-7\). Upon entering this function, the graphing utility depicts the shape of the curve immediately on the screen. You can observe how quickly the function grows for positive values and how it flattens and approaches the x-axis for negative values of \(x\). This visualisation is crucial for understanding the behavior and properties of the function beyond analytical computations.Beyond static plotting, many graphing utilities allow you to zoom in and out, providing a closer look at specific regions of the graph or observing overall trends. This feature enables a greater comprehension of how minor modifications in the function affect its graph.

The Art of Inequality Shading

Inequality shading is a technique used to represent the solution set of an inequality on a graph. This involves coloring a region on the graph where all points satisfy the inequality condition.With the given inequality \(y \leq 2^{2x-0.5}-7\), the task is to visualize all y-values less than or equal to the function. Start by plotting the function \(f(x) = 2^{2x-0.5}-7\) using your graphing utility. Once the curve is displayed, target the space directly under this curve for shading.

• All points on the curve are included (as indicated by the '\(\leq\)' symbol), finalizing this as a solid boundary.
• Shade the region beneath the curve, extending this space indefinitely until the bounds of the graph.

By shading this area, you demonstrate all points \((x, y)\) where \(y\) is less than or equal to the function's output, effectively showing the solution set of the inequality. This method transforms abstract inequality statements into visual insights, enabling deeper understanding and analysis.