Problem 20 - College Algebra

Question

For the following exercises, find the formula for an exponential function that passes through the two points given. $$ \left(-1, \frac{3}{2}\right) \text { and }(3,24) $$

Step-by-Step Solution

Answer

The formula for the exponential function is \( y = 3 \times 2^x \).

Step 1: Understanding the Exponential Function

An exponential function can be expressed as \( y = ab^x \), where \( a \) is the initial value and \( b \) is the growth rate. We will use this form to find the values of \( a \) and \( b \) using the given points.

Step 2: Substituting the First Point

Substitute the coordinates of the first point \((-1, \frac{3}{2})\) into the equation: \( \frac{3}{2} = ab^{-1} \), which simplifies to \( \frac{3}{2} = \frac{a}{b} \).

Step 3: Substituting the Second Point

Substitute the coordinates of the second point \((3, 24)\) into the equation: \( 24 = ab^3 \).

Step 4: Solving the System of Equations

We have two equations: \( \frac{3}{2} = \frac{a}{b} \) and \( 24 = ab^3 \). Solve the first equation for \( a \): \( a = \frac{3}{2}b \). Substitute \( a \) into the second equation: \( 24 = \left(\frac{3}{2}b\right)b^3 = \frac{3}{2}b^4 \).

Step 5: Finding the Value of \( b \)

Solve the equation \( 24 = \frac{3}{2}b^4 \) by multiplying both sides by 2: \( 48 = 3b^4 \) and then divide by 3: \( b^4 = 16 \). Now take the fourth root: \( b = 2 \).

Step 6: Finding the Value of \( a \)

Using \( a = \frac{3}{2}b \) and substituting \( b = 2 \), we find \( a = \frac{3}{2} \times 2 = 3 \).

Step 7: Writing the Exponential Function

Now that we have \( a \) and \( b \), substitute these into the exponential equation: \( y = 3 \times 2^x \).

Key Concepts

System of Equations

When working with exponential functions, you might need to solve a system of equations to find unknown parameters. In this context, a system of equations refers to a set of two or more equations that use the same variables. Solving it means finding the values of those variables that satisfy all equations simultaneously. In our problem, we have determined two key equations from the points:

• \( \frac{3}{2} = \frac{a}{b} \)
• \( 24 = ab^3 \)

These equations represent our 'system'. To solve for \( a \) and \( b \), we took the following steps:- From the first equation, we solved for one variable in terms of the other: \( a = \frac{3}{2}b \).- Next, we substituted \( a = \frac{3}{2}b \) into the second equation.By doing this, we managed to reduce the two equations down to a single equation with one variable, making it easier to find the solution. This kind of method is essential when working with systems of equations in exponential functions.

Exponential Growth

Exponential growth occurs when the rate of change in a quantity is proportional to the current amount of that quantity. In simpler terms, it's when something grows by the same multiplicative factor over equal increments of time. This is fundamentally different from linear growth, where the amount of change per unit time is constant.In an exponential function like \( y = ab^x \):

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

In our specific example, we found \( b = 2 \), indicating that every increase by one unit in \( x \) will result in the output value doubling. This rapid increase exemplifies exponential growth effectively, and understanding this concept helps in interpreting problems involving processes such as population growth, radioactive decay, or interest compounds.

Function Formula

The formula of a function is a mathematical rule that assigns each input to a single output. For exponential functions like the one we are dealing with, the formula is expressed as \( y = ab^x \), where:

• \( a \) is the initial or starting value, which is the value of \( y \) when \( x \) is zero.
• \( b \) is the base or growth factor, determining the rate of increase or decrease.

When looking at our exercise, after determining the values \( a = 3 \) and \( b = 2 \), we were able to write the specific formula for the exponential function as \( y = 3 \times 2^x \).This formula can now be used to calculate the output for any given input. It is crucial to understand how each part of the formula affects the behavior of the function. For instance, changing \( a \) would shift the graph up or down, while changing \( b \) alters the growth rate. Learning how to construct these function formulas is vital for solving numerous real-world problems, making them a fundamental concept in mathematics.