Problem 20 - College Algebra with Corequisite Support

Question

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

Step-by-Step Solution

Answer

The formula is \( y = 3 \cdot 2^x \).

Step 1: Understand the Exponential Function

An exponential function is generally expressed as \( y = ab^x \), where \( a \) is the initial value (when \( x = 0 \)) and \( b \) is the base or growth factor of the function.

Step 2: Substitute the First Point

We have the point \( \left(-1, \frac{3}{2}\right) \). Substitute this into the equation: \[ \frac{3}{2} = ab^{-1} \]This implies that \( ab^{-1} = \frac{3}{2} \) or \( \frac{a}{b} = \frac{3}{2} \).

Step 3: Substitute the Second Point

We also have the point (3, 24). Substitute this into the equation:\[ 24 = ab^3 \]

Step 4: Solve the System of Equations

From Step 2, we have \( a = \frac{3}{2}b \) and from Step 3, \( ab^3 = 24 \). Substitute \( a = \frac{3}{2}b \) into the second equation:\[ \left(\frac{3}{2}b\right)b^3 = 24 \]\[ \frac{3}{2}b^4 = 24 \]

Step 5: Solve for the Base b

Solve the equation \( \frac{3}{2}b^4 = 24 \):\[ b^4 = \frac{24 \times 2}{3} \]\[ b^4 = 16 \]\[ b = 16^{\frac{1}{4}} \]\[ b = 2 \]

Step 6: Solve for the Initial Value a

Using \( b = 2 \) in \( a = \frac{3}{2}b \):\[ a = \frac{3}{2} \times 2 \]\[ a = 3 \]

Step 7: Write the Exponential Function

Substitute \( a = 3 \) and \( b = 2 \) into the general formula:\[ y = ab^x = 3 \cdot 2^x \]

Key Concepts

Points on a Graph

When we talk about points on a graph in relation to exponential functions, we're essentially referring to specific coordinates that the function goes through. These points are presented as pairs

1. where the first number is the x-coordinate
2. and the second number is the corresponding y-coordinate.

For example, the points

1. \((-1, \frac{3}{2})\)
2. and \((3, 24)\)

indicate that when

• \(x = -1\), \(y\) or the function value is \(\frac{3}{2}\),
• and when \(x = 3\), \(y = 24\).

In exponential functions, such points help us determine critical parameters like the initial value and the growth factor.
By plotting these points on a Cartesian plane, students can visualize the shape and direction of the exponential graph, typically characterized by rapid increase or decrease. The challenge lies in identifying or confirming the function that connects these points.

System of Equations

A system of equations comes into play when we need to solve for unknown variables using two or more equations. In the context of exponential functions, it's how we use the given points to find the equation connecting them.
With two equations formed from the points

• \((-1, \frac{3}{2})\)
• and \((3, 24)\),

a student would typically substitute these into the exponential equation form \(y = ab^x\). This process generates a set of equations:

• From the point \((-1, \frac{3}{2})\), becomes \(\frac{3}{2} = \frac{a}{b}\).
• From the point \((3, 24)\), becomes \(24 = ab^3\).

These two equations are solved together as a system to find \(a\) and \(b\). By substituting and simplifying step by step, students learn logical problem-solving skills, beginning with calculating \(b\) and then finding \(a\).
The systematic approach provides clarity on how exponential functions are defined by unique characteristics like growth rates and initial values.

Growth Factor

The growth factor in an exponential function is a pivotal component that depicts how the function's value changes with each step in the x-direction. In the formula \(y = ab^x\):

• \(b\) represents the growth factor.

This tells us whether the function is increasing or decreasing as \(x\) increases.
For instance, in our problem where \(b = 2\), this indicates that the function value doubles every time \(x\) increases by one unit. If \(b\) were less than 1, it would mean the function is decreasing, showing decay.
Understanding this concept is crucial for recognizing patterns in data, modeling exponential growth scenarios like population growth, or decay processes like radioactive decay. By calculating \(b = 2\) from the system of equations, students see firsthand how real-world phenomena can be quantitatively described and predicted using exponential models.