Problem 64 - Algebra 2

Question

Write an exponential equation in the form \(y=a b^{x}\) for a graph that includes \((2,54)\) and \(\left(\frac{1}{2}, 2\right) .\) Show your work.

Step-by-Step Solution

Answer

The exponential equation that passes through the points (2,54) and (0.5, 2) is \(y=\frac{2}{3} \times 9^{x}\).

Step 1: Substitute the given points into the generic exponential function

The generic form of an exponential function is given by \(y=a b^{x}\). Substitute the given points \((2,54)\) and \(\left(\frac{1}{2}, 2\right)\) into the equation. This will create a system of two equations which can be written as:\n1. Substitute \((2,54)\) into the equation, we get \(54=a b^{2}\) (Eqn 1).\n2. When we put \(\left(\frac{1}{2}, 2\right)\) into the equation we get \(2=a b^{\left(\frac{1}{2}\right)}\) (Eqn 2).

Step 2: Solve the System of Equations

The system of equations could be solved by first isolating 'a' from Eqn 2: \(a=\frac{2}{b^{\left(\frac{1}{2}\right)}}\).\nSubstitute this value of 'a' in equation 1, we get \(54=\frac{2}{b^{\left(\frac{1}{2}\right)}} \times b^{2}\).\nOr, \(54=2 \times b^{\left(\frac{3}{2}\right)}\).\nSo, \(b^{\left(\frac{3}{2}\right)}=27\).\nTo solve for 'b', raise each side of the equation to the power of \(\frac{2}{3}\) in order to get \(b=\left(27\right)^{\frac{2}{3}}=9\)

Step 3: Find the Value of coefficient 'a'

Substitute the value of 'b' in Eqn 2: \(2=a \times 9^{\left(\frac{1}{2}\right)} \rightarrow 2=a \times 3\).\nDivide each side by '3' to get the value of 'a', \(\therefore a=\frac{2}{3}.\)

Step 4: Formulate the Exponential Equation

Substitute values of 'a' and 'b' into the general form of the equation. The equation would look like: \(y=\frac{2}{3} \times 9^{x}\)

Key Concepts

Exponential Functions

Exponential functions are mathematical expressions that model real-world phenomena where rapid change is a central feature. These functions have the form \(y = a b^{x}\). Here, "\(a\)" represents the initial amount, "\(b\)" denotes the base of the exponential function, and "\(x\)" is the exponent or the rate of change. These functions are known for their characteristic rapid increase or decrease, depicted graphically with a curve that either rises or falls swiftly depending on the value of \(b\).

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

Understanding how to work with exponential functions is critical in various fields, such as finance for calculating compound interest, population biology for modeling growth in populations, and physics for radioactive decay analysis.

Solving Equations

Solving equations is a fundamental skill in mathematics that involves finding the value of unknowns that make the equation true. For exponential equations, this often involves working with powers and understanding expressions with exponents. The key to solving these kinds of problems is to manipulate the equation into a simpler form using algebraic principles.
In the case of solving the exponential equation form \(y = a b^{x}\), vital steps include:

• Substituting known values into the equation to create a system of equations if necessary.
• Isolating variables through algebraic manipulation to find their specific values.
• Ensuring calculations are accurate, especially with exponents which follow specific rules.

These techniques are useful in making complex problems more approachable and finding exact solutions.

Mathematical Modeling

Mathematical modeling uses mathematical expressions to represent real-world systems. Exponential functions are prevalent in mathematical modeling due to their natural occurrence in processes involving growth and decay. For instance, understanding how bacteria populations grow or how investments increase over time helps predict future behaviors based on current trends.
Mathematical models aid in:

• Interpreting and simplifying complex systems.
• Making predictions about future events.
• Testing hypotheses and validating theoretical concepts.

Models are essential tools for supporting decision-making, planning, and strategic forecasting in various disciplines, highlighting the power of mathematics in practical applications.

System of Equations

A system of equations consists of multiple equations that need to be solved together since they share variables. In exponential problems, you often create a system by substituting known data points into the function. This method allows you to derive specific values for the constants involved, like "\(a\)" and "\(b\)" in an exponential equation.
For instance, from two data points, you formulate two different equations, representing them as a system. Solving it involves methods such as:

• Substitution: Solving one equation for a variable and substituting the result in another.
• Elimination: Adding or subtracting equations to eliminate a variable.
• Matrix methods: Applying linear algebra techniques for more complex systems.

Understanding systems of equations is invaluable for understanding how different variables interact within a mathematical model, providing deeper insights into complex problems.