Problem 54 - Precalculus

Question

For the following exercises, use a graphing calculator to find the equation of an exponential function given the points on the curve. $$ (5,2.909) \text { and }(13,0.005) $$

Step-by-Step Solution

Answer

The exponential equation is \( y = 119.49(0.3435)^x \).

Step 1: Understand the Exponential Function

To solve for the equation of an exponential function, we begin by recalling that exponential functions have the form \( y = ab^x \). Our task is to determine the values of \( a \) and \( b \) using the given points.

Step 2: Set Up the System of Equations

Using the points \((5, 2.909)\) and \((13, 0.005)\), we can set up two equations based on the form \( y = ab^x \). So we have:\[ 2.909 = ab^5 \]\[ 0.005 = ab^{13} \]

Step 3: Solve for \( b \)

To eliminate \( a \), divide the second equation by the first:\[ \frac{0.005}{2.909} = \frac{ab^{13}}{ab^5} \Rightarrow b^{8} = \frac{0.005}{2.909} \]Calculate \( b \):\[ b = \left( \frac{0.005}{2.909} \right)^{\frac{1}{8}} \approx 0.3435 \]

Step 4: Solve for \( a \)

Substitute \( b \) back into one of the original equations to solve for \( a \). Using \( ab^5 = 2.909 \):\[ a(0.3435)^5 = 2.909 \]\[ a = \frac{2.909}{(0.3435)^5} \approx 119.49 \]

Step 5: Write the Final Equation

Now that we have \( a \) and \( b \), substitute them into the exponential function form to get:\[ y = 119.49(0.3435)^x \]

Key Concepts

Graphing Calculator

A graphing calculator is an indispensable tool when working with functions, especially exponential ones. These calculators allow you to input equations and visualize their graphs, making it easier to understand complex relationships and patterns. To use a graphing calculator for finding an exponential equation, start by inputting your known data points.
This involves setting up the points on a coordinate grid.

• Input the coordinates, such as \((5, 2.909)\) and \((13, 0.005)\).
• Plot these points on the calculator's graphing interface.
• Use the calculator's feature to fit an exponential curve to your data.

Most graphing calculators will have a function to determine the best-fit line or curve, be it linear, polynomial, or exponential. By leveraging this function, you can find the values of the coefficients in the equation \( y = ab^x \).
This equation describes the behavior of the curve that best fits your data points.

System of Equations

A system of equations is a set of equations with multiple variables that you solve together. When finding an exponential equation from given points, you can set up a system of equations to solve for unknown coefficients.
For instance, with the points \((5, 2.909)\) and \((13, 0.005)\), we create two equations in the form:

• \( 2.909 = ab^5 \)
• \( 0.005 = ab^{13} \)

The goal is to solve for \( a \) and \( b \).
By dividing these equations, you can eliminate one variable to ease solving for the other. This step reduces complexity and helps find the base \( b \).
Once \( b \) is found, substitute it back to find the initial amount \( a \).
Solving systems of equations is crucial in determining precise equations representing exponential behavior.

Exponential Growth and Decay

Exponential functions can model both growth and decay phenomena in various disciplines, from biology to finance. These functions are expressed in the form \( y = ab^x \), where \( a \) is the initial value, and \( b \) is the growth (or decay) factor.

• If \( b > 1 \), the function represents exponential growth.
• If \( 0 < b < 1 \), it represents exponential decay.

For the solution provided, we discovered that \( b \approx 0.3435 \); this indicates an exponential decay because \( b \) is less than 1.
Understanding whether a function shows growth or decay helps interpret the real-world phenomena it models, like population changes or radioactive decay.Conceptually, exponential processes involve repeated multiplication. In exponential decay, each step reduces the original quantity by a constant proportion, visualizable through the graphing calculator's curve plotted from the equation \( y = 119.49(0.3435)^x \).
This comprehension of growth and decay is essential in solving problems and making predictions based on real-world data.