Problem 46 - College Algebra with Corequisite Support

Question

For the following exercises, use a graphing calculator to approximate the solutions of the equation. Round to the nearest thousandth. \(-50=-\left(\frac{1}{2}\right)^{-x}\)

Step-by-Step Solution

Answer

The approximate solution is \(x \approx 5.646\).

Step 1: Rewrite the Equation

Start by rewriting the given equation \(-50=-\left(\frac{1}{2}\right)^{-x}\).Realize that we need to set up the equation for graphing. To make it easier, we'll rewrite this as \(y = -\left(\frac{1}{2}\right)^{-x} + 50 = 0\).

Step 2: Graph the Equation

On a graphing calculator, input the equation \(y = -\left(\frac{1}{2}\right)^{-x} + 50\).Look for the x-value(s) where the graph intersects the x-axis. This will give you the solution(s) for the equation.

Step 3: Approximate the Solution

Use the graph to find the x-intercept(s), which is the point(s) where the graph crosses the x-axis. Approximating this point using the calculator's root-finding feature, find the value for \(x\) such that \(-\left(\frac{1}{2}\right)^{-x} + 50 = 0\).

Step 4: Round to the Nearest Thousandth

Once you have the approximate x-value from the graph, round the solution to the nearest thousandth to get your final answer. Suppose the x-intercept you found was 5.646, rounding to the nearest thousandth gives 5.646.

Key Concepts

Exponential Equations

Exponential equations are mathematical expressions where variables appear as exponents. These equations are integral in showing rapid growth or decay, such as in population models or radioactive decay processes. Generally, an exponential equation can look like

• where the base (in our case, \(\frac{1}{2}\)) is a constant.
• The variable is found in the exponent position.

To solve exponential equations, you can use algebraic techniques to transform them into a graphable format. For example, rewriting the original problem as \(y = -\left(\frac{1}{2}\right)^{-x} + 50 = 0\) allows for graphing, making it easier to visualize the behavior of the equation.
When dealing with these types of equations on a graphing calculator, your goal is to find where the function intersects the x-axis. Understanding and accurately graphing these functions is crucial since it provides a visual representation of potential solutions.

X-Intercept

Finding the x-intercept of an equation is a key step in solving it graphically. The x-intercept refers to the point where the graph crosses the x-axis, which means the output value is zero. Essentially, this is the solution to the equation.
In the context of our given equation \(y = -\left(\frac{1}{2}\right)^{-x} + 50\), identifying the x-intercept involves finding when the value of \(y\) equals zero.
Using a graphing calculator efficiently:

• Enter the expression \(-\left(\frac{1}{2}\right)^{-x} + 50\) into the calculator.
• Highlight the intersection point on the x-axis with the calculator’s root-finding tool, which provides the x-value(s) for which the function is zero.

Determining the x-intercept is essential because it signifies the possible solutions where the effects of exponential growth or decay are balanced out by the constant.

Approximating Solutions

Approximating solutions is often necessary when the exact answer isn't easily obtainable due to the equation's complexity or irrational nature. With the help of technology like graphing calculators, you can find solutions to a high degree of accuracy without solving analytically.
Start by visualizing the function on the graph to see where it crosses the x-axis. The x-value(s) at this point are your approximate solutions.
Here’s how to approximate solutions:

• Input the equation into your graphing calculator and graph it.
• Use the calculator's features, such as the "trace" or "root" tool, to pinpoint the x-intercept.
• Note this x-value and ensure accuracy by adjusting the viewing window if needed.
• Lastly, round the solution to the desired precision—in this case, the nearest thousandth.

Rounding ensures that the solution is practical for real-world application. For instance, if the graph predicts an x-value like 5.6458, rounding it to 5.646 offers an efficient and usable answer. Approximating through graphical methods is invaluable in making complex mathematical tasks manageable.