Problem 9 - Algebra 2

Question

Write an exponential function for the graph that passes through the given points. $$ (0,-18) \text { and }(-2,-2) $$

Step-by-Step Solution

Answer

The exponential function is \( f(x) = -18 \, \times \, 3^x \).

Step 1: Understand the Form of an Exponential Function

An exponential function can be generally expressed as \( f(x) = ab^x \) where \( a \) is the initial value and \( b \) is the base or growth/decay factor.

Step 2: Use the Point (0, -18) to Find Parameter a

Substituting the point \( (0, -18) \) into the exponential function form, we get:\[ f(0) = ab^0 = a = -18 \]Hence, \( a = -18 \).

Step 3: Substitute a into the Exponential Function

With \( a = -18 \), the function becomes \( f(x) = -18b^x \).

Step 4: Use the Point (-2, -2) to Find Parameter b

Substitute \( x = -2 \) and \( f(x) = -2 \) into \( f(x) = -18b^x \):\[ -2 = -18b^{-2} \]This rearranges to:\[ b^{-2} = \frac{-2}{-18} = \frac{1}{9} \]

Step 5: Solve for Parameter b

Take the reciprocal of both sides of the equation to solve for \( b^2 \):\[ b^2 = 9 \]Taking the square root of both sides, we find:\[ b = 3 \text{ (since \( b\) is positive in most contexts) } \]

Step 6: Write the Final Exponential Function

Substitute \( b = 3 \) back into the function \( f(x) = -18b^x \):\[ f(x) = -18 \, \times \, 3^x \]. This is the exponential function that passes through the given points.

Key Concepts

Growth/Decay Factor

In exponential functions, the growth or decay factor plays a critical role in determining how the function behaves. The function is typically written as \( f(x) = ab^x \), where \( b \) is the growth or decay factor. When \( b \) is greater than 1, the function represents exponential growth, indicating that the function values increase rapidly. Conversely, when \( 0 < b < 1 \), the function depicts exponential decay, meaning the values decrease over time.
In the given example, we found that \( b = 3 \). Since 3 is greater than 1, this indicates that the function reflects exponential growth. Understanding whether \( b \) involves growth or decay helps us anticipate the behavior of the graph, making it an essential concept of exponential functions.

Initial Value

The initial value in an exponential function refers to the value of the function at \( x = 0 \). This is denoted by \( a \) in the function \( f(x) = ab^x \). It represents the starting point or the y-intercept of the graph of the function. Essentially, it indicates where the graph intersects the y-axis. Having a firm grasp on the initial value is fundamental, as it sets a reference point for how the function will expand or contract with changes in \( x \).
In the exercise, the point \((0, -18)\) was used to determine the initial value. By substituting this point into the equation, we found that \( a = -18 \). This implies that the exponential function begins at \( -18 \) when \( x \) is zero, providing a baseline for further calculations and graphing.

Exponential Equation Solving

Solving exponential equations involves manipulating the equation to find unknown values, like the growth/decay factor or other parameters. Typically, this requires substituting given points into the exponential function and solving for the unknowns.
In our example, we initially determined the parameter \( a \) using point \((0, -18)\). Then we substituted point \((-2, -2)\) into the function \( f(x) = -18b^x \) to solve for \( b \). From the equation \( -2 = -18b^{-2} \), we simplified to find \( b^{-2} = \frac{1}{9} \). By taking the reciprocal and the square root, we found \( b = 3 \).
This approach demonstrates the structured steps of solving exponential equations: identifying known values, substituting them into the function, and solving for the unknowns through algebraic transformations. Mastery of this process is essential for successfully working with exponential functions.