Problem 22 - College Algebra with Corequisite Support

Question

For the following exercises, find the formula for an exponential function that passes through the two points given. (3,1) and (5,4)

Step-by-Step Solution

Answer

The function is \( y = \frac{1}{8} imes 2^x \).

Step 1: Understanding the Problem

We need to find an exponential function of the form \( y = ab^x \) that passes through the points (3,1) and (5,4). This means the function should produce these y-values for the given x-values.

Step 2: Set Up the Equations

Substitute the points into the exponential function form. For the point (3,1): \[ 1 = ab^3 \]For the point (5,4): \[ 4 = ab^5 \].

Step 3: Divide the Equations to Solve for b

Divide the second equation by the first to eliminate \(a\):\[ \frac{4}{1} = \frac{ab^5}{ab^3} \]This simplifies to:\[ 4 = b^2 \].

Step 4: Solve for b

Take the square root on both sides to solve for \(b\):\[ b = 2 \].

Step 5: Substitute b back to find a

Substitute \(b = 2\) back into the first equation:\[ 1 = a(2^3) \]\[ 1 = a imes 8 \]\[ a = \frac{1}{8} \].

Step 6: Write the Exponential Function

With values \(a = \frac{1}{8}\) and \(b = 2\), the exponential function is:\[ y = \frac{1}{8} imes 2^x \].

Key Concepts

Algebra

Algebra is a powerful branch of mathematics dealing with symbols and the rules for manipulating those symbols. Whenever we deal with variables like \( x \) and \( y \), and form expressions and equations with them, we are basically doing algebra. It allows us to understand relationships between numbers and define mathematical operations in a concise manner.

• **Variables**: Think of them as placeholders for numbers. In our problem, \( x \) and \( y \) represent input and output values, respectively.
• **Constants**: These are numbers that have fixed values. In our formula \( y = ab^x \), \( a \) and \( b \) are constants once we figure them out through the provided points.
• **Expressions**: Combinations of variables and constants using mathematical operations. The right side of the equation, \( y = ab^x \), is itself an expression.

When we set up the equations \( 1 = ab^3 \) and \( 4 = ab^5 \) using the points (3,1) and (5,4), we are expressing a particular relationship between \( x \) and \( y \) that needs to be solved. Understanding how these expressions relate to algebraic equations paves the way to solving the entire problem.

Mathematical Equations

Mathematical equations are statements that assert the equality of two expressions. In the context of our problem, each of the given points helps us to form an equation from the exponential function form \( y = ab^x \).

• **Substitution**: This involves replacing variables with specific values to find unknowns. We substituted point values into our general equations, \( 1 = ab^3 \) and \( 4 = ab^5 \), leading to specific equations.
• **Manipulation**: Manipulating equations can mean adding, subtracting, multiplying, or dividing both sides to solve for variables. When we divided \( \frac{4}{1} = \frac{ab^5}{ab^3} \), it allowed us to isolate and simplify to find \( b \).
• **Simplifying**: This simplifies the terms and helps in solving complex expressions. It was crucial for reaching the step \( 4 = b^2 \), leading us to solve for one of the unknowns.

Understanding mathematical equations and knowing how to form and manipulate them are vital skills when it comes to finding the values of unknown variables in complex mathematical problems.

Function Solving

Solving a function means finding a relationship between input (\( x \)) and output (\( y \)) values that satisfy the function given specific conditions. With exponential functions, these relationships can model various real-world situations, from population growth to radioactive decay.

• **Finding Parameters**: In solving for \( a \) and \( b \) in \( y = ab^x \), we step through using our derived equations, ultimately finding each parameter. With \( b = 2 \) from \( 4 = b^2 \), we backtrack into the initial equation to find that \( a = \frac{1}{8} \).
• **Exponentials**: Recognizing the form and significance of an exponential function is critical. The base \( b \) controls the growth rate, and the coefficient \( a \) scales the output.
• **Constructing the Function**: With our parameters solved, the function \( y = \frac{1}{8} \times 2^x \) is a concise representation of the relationship that aligns with the points (3,1) and (5,4).

Solving functions involves understanding these parameters and knowing how to adjust them so the function behaves as expected for given data points. With practice, this becomes an intuitive process for modeling and solving various mathematical and real-life problems.