Problem 66 - Algebra 2

Question

Write an exponential equation \(y=a b^{x}\) whose graph passes through the given points. \((1,6)\) and \((2,12)\)

Step-by-Step Solution

Answer

The exponential equation whose graph passes through the points \((1,6)\) and \((2,12)\) is \(y = 3*2^x\).

Step 1: Formulate the first equation using the first point

We can substitute the first point \((1,6)\) into the equation \(y = ab^x\) to get \(6 = ab^1\) which can be simplified to \(6 = ab\).

Step 2: Formulate the second equation using the second point

We now substitute the second point \((2,12)\) into the equation \(y = ab^x\) to get \(12 = ab^2\).

Step 3: Solve for the variable \(b\)

To find the value of \(b\), we can divide the equation from Step 2 by the equation from Step 1. This gives us: \(\frac{12}{6} = \frac{ab^2}{ab}\), which simplifies to \(2 = b\). Therefore, \(b = 2\).

Step 4: Substitute \(b\) into the equation from Step 1 to find \(a\)

Now that we have found \(b = 2\), we can substitute this into the equation from Step 1 to solve for \(a\). This gives \(6 = a * 2\), which simplifies to \(a = 3\).

Key Concepts

Algebra 2

Algebra 2 is a critical step in high school mathematics, where students build on the principles learned in Algebra 1. This subject introduces more complex equations, including exponential equations like the one in the problem. Understanding Algebra 2 involves studying how different types of equations work and how to manipulate them to find solutions. Students explore the properties of numbers and learn how to express real-world situations as mathematical equations.

In the context of the original exercise, we use an exponential equation, which is a type of function that can model growth or decay that happens at increasing rates. Here, the task is to find the specific equation that fits the two points given. Such problems aim to develop critical thinking and equation-solving skills, which are key components of Algebra 2.

Solving Equations

Solving equations is a central skill in Algebra 2 that requires isolating a variable to determine its value. In the original exercise, we start with an exponential equation of the form \( y = ab^x \) and use given points to create two separate equations.

The first step involves substituting the first set of coordinates \((1,6)\) into the equation, which gives \(6 = ab\). Similarly, the second set of coordinates \((2,12)\) helps form the equation \(12 = ab^2\). By having two equations, we can manipulate them to solve for the unknowns \(a\) and \(b\).

• First, division of the second equation by the first allows us to eliminate \(a\) and simplify the exponential part, ultimately solving for \(b\).
• Next, we use the found value of \(b\) to substitute back into one of the initial equations to solve for \(a\).

This step-by-step method is essential in solving more complex equations as students advance in their understanding of algebraic concepts.

Exponential Functions

Exponential functions represent scenarios where growth or decay is proportional to the current value. These functions are expressed in the format \( y = ab^x \). The task in the original exercise involved determining the constants \(a\) and \(b\) for these specific conditions.

Exponential functions can describe a wide range of real-world phenomena, from population growth to radioactive decay. They are unique because the change increases or decreases multiplicatively, often leading to quicker and more dramatic shifts compared to linear functions.

To solve the original problem, we identified the values of \(a\) and \(b\) as follows:

• By dividing the equations derived from the two given points, we isolated \(b\).
• Subsequently, with the known \(b\), we went back to determine \(a\).

The final equation, \( y = 3(2)^x \), beautifully illustrates how exponential functions can model rapidly changing situations by utilizing key mathematical principles.