Problem 15 - Algebra 2

Question

Write an exponential function \(y=a b^{x}\) for a graph that includes the given points. $$ (0,24),\left(3, \frac{8}{9}\right) $$

Step-by-Step Solution

Answer

The exponential function that goes through the points (0, 24) and (3, 8/9) is \(y = 24 \cdot (\frac{1}{2})^{x}\).

Step 1: Find the initial value \(a\)

The first point given is (0,24), which means when \(x = 0\), \(y = 24\). Now, set \(x = 0\) and \(y = 24\) in the general formula \(y = a \cdot b^{x}\). This simplifies to \(24 = a \cdot b^{0}\), and any number raised to the power of 0 is 1. So, the initial value \(a\) is 24.

Step 2: Find the base \(b\)

The second point given is \(3, \frac{8}{9}\), which means when \(x = 3\), \(y = \frac{8}{9}\). Substitute the values of \(a\), \(x\), and \(y\) into the general equation \(y = a \cdot b^{x}\). This gives us \(\frac{8}{9} = 24 \cdot b^{3}\). Solving this equation for \(b\), you would see that \(b\) is equal to \(\frac{1}{2}\).

Step 3: Assemble the function

With the values of \(a\) and \(b\), we substitute into the general function \(y = a \cdot b^{x}\), leading to the final function \(y = 24 \cdot (\frac{1}{2})^{x}\).

Key Concepts

Graphing Exponential Functions

When graphing exponential functions, it’s key to understand how the graph of the function behaves under various conditions. An exponential function generally takes the form \(y = ab^x\), where \(a\) is the initial value, and \(b\) is the base of the exponential function. The graph will show how the value of \(y\) changes as \(x\) increases or decreases. If \(b > 1\), the graph shows exponential growth, with \(y\) increasing rapidly as \(x\) increases. Conversely, if \(0 < b < 1\), the graph depicts exponential decay, meaning \(y\) gets smaller as \(x\) increases. Let's illustrate this:

• If \(b = 2\), the graph doubles each step to the right.
• If \(b = \frac{1}{2}\), the graph halves each step to the right.

Graphs of exponential functions often appear in various practical applications, such as modeling population growth, radioactive decay, and interest calculations.

Finding Initial Value

The initial value of an exponential function is the value of \(y\) when \(x = 0\). This value is vital as it sets the starting point for how the graph behaves. To find the initial value \(a\), substitute \(x = 0\) into the exponential function equation \(y = ab^x\). This simplifies to \(y = a\). For example, looking at the point \((0, 24)\) given in the problem, since \(x = 0\), it directly tells us that \(a = 24\). It's a straightforward step where you essentially get \(a\) directly from the value of \(y\) at \(x = 0\). You're identifying the starting height of your graph on the \(y\)-axis.

Determining Base of Exponential

Finding the base of an exponential function tells us how the function will behave - whether it will grow or decay. To determine the base \(b\), select another point on the graph, substitute the \(x\) and \(y\) values into the equation, keeping the initial value \(a\) you already found. Simplify and solve the equation for \(b\). In this exercise, using the point \((3, \frac{8}{9})\), plug it into the equation as follows: \(\frac{8}{9} = 24 \cdot b^3\). By solving for \(b\), one might divide to isolate \(b^3\), resulting in \(b = \frac{1}{2}\). Knowing \(b = \frac{1}{2}\) means each time \(x\) increases by 1, \(y\) is halved, indicating a decay situation.

Solving Exponential Equations

Solving exponential equations involves isolating the exponential term to find the unknown variable, often the exponent or the base. After gathering your initial and base values, check consistency with the function by plugging in other points. By taking the function \(y = 24 \cdot (\frac{1}{2})^x\), if you have further equations, substitute \(x\) values to see if \(y\) matches expected results:

• For \((3, \frac{8}{9})\), when \(x = 3\), plug into \(24 \cdot (\frac{1}{2})^3 = \frac{8}{9}\) to verify consistency.
• Continue to substitute other values as needed to test predictions.

This approach allows you to see the result of inputting different \(x\) values, reinforcing your understanding by practice and confirmation.