Problem 64 - A Graphical Approach to Precalculus with Limits

Question

Give an exponential function in the form \(f(x)=a^{x}\) whose graph contains the given point. (a) \((3,8)\) (b) \((-3,64)\)

Step-by-Step Solution

Answer

(a) \( f(x) = 2^x \); (b) \( f(x) = \left(\frac{1}{4}\right)^x \)."

Step 1: Understand the equation form

We are looking for an exponential function of the form \(f(x) = a^x\). In this function, taking any point \((x, y)\), \(y\) should equal \(a^x\). Therefore, we need to solve for \(a\) using the given point.

Step 2: Solve for (a) using the given point (3, 8)

We are given the point \((3, 8)\). This means that when \(x = 3\), \(y = f(x) = 8\). Substitute these into the exponential equation, \(a^3 = 8\). To find \(a\), take the cube root of both sides: \(a = 8^{1/3} = 2\). So, the function is \(f(x) = 2^x\).

Step 3: Solve for (b) using the given point (-3, 64)

We are given the point \((-3, 64)\). At \(x = -3\), \(y = f(x) = 64\). Substitute into the equation: \(a^{-3} = 64\). Rewriting \(a^{-3}\) as \(\frac{1}{a^3}\), we have \(\frac{1}{a^3} = 64\), implying \(a^3 = \frac{1}{64}\). Thus, \(a = \left(\frac{1}{64}\right)^{1/3} = \frac{1}{4}\). Therefore, the function is \(f(x) = \left(\frac{1}{4}\right)^x\).

Key Concepts

Graphing Exponential Functions

When it comes to understanding exponential functions, graphing can offer a visual perspective that makes comprehension simpler. The general form of an exponential function is \(f(x) = a^x\), where \(a\) is a constant referred to as the base, and \(x\) is the exponent.

• If \(a > 1\), the graph of the exponential function rises exponentially, which means it grows rapidly as \(x\) increases.
• If \(0 < a < 1\), the graph decreases exponentially, shrinking towards zero as \(x\) grows.

For instance, consider a point like (3, 8) on the graph of such a function. This point helps us identify the specific function's characteristics by allowing us to reverse-engineer the value of \(a\). Knowing one point, we can better predict and sketch the entire behavior of the function. Think of drawing out a curve that passes through this point, indicating whether the function is rapidly increasing or decreasing across the x-axis.
Drawing these curves effectively requires choosing several x-values to calculate the corresponding y-values and plotting these on a graph to observe the function’s growth or decay pattern.

Solving Exponential Equations

Finding the actual function behind an exponential expression involves solving an equation where you typically know a point \((x, y)\) and need to determine the base \(a\).
For example, if we know a point (3, 8) belongs to an exponential function, we write \(8 = a^3\).

• To solve for \(a\), simply take the relevant root of \(8\), which is \(8^{1/3} = 2\).
• For the point (-3, 64), the equation becomes \(64 = a^{-3}\). Since \(a^{-3} = \frac{1}{a^3}\), we then solve \(a^3 = \frac{1}{64}\), which yields \(a = \left(\frac{1}{64}\right)^{1/3} = \frac{1}{4}\).

This problem-solving process enables us to arrive at specific values for \(a\), leading us to the exponential function’s formula. These solved equations showcase a function that either grows or decays across its entire domain, depending on whether the base \(a\) is greater than or less than one.

Function Representation

Understanding how exponential functions are represented is crucial to working with them effectively. The function \(f(x) = a^x\) represents an equation where the output, \(f(x)\), is the result of raising the base \(a\) to the power of \(x\). This results in a distinctive curve on a graph.

• The base \(a\) acts as the primary influence on the function's growth or decay.
• If \(a > 1\), the representation is an increasing function, illustrating rapid growth with increasing \(x\).
• If \(0 < a < 1\), the function is decreasing, indicating exponential decay where the values drop as \(x\) increases.

In practice, converting a specific point, like (3, 8), into the function \(f(x) = 2^x\) means acknowledging that for each unit increase in \(x\), the value of \(f(x)\) is multiplied by 2. Conversely, with the point (-3, 64), the corresponding function \(f(x) = \left(\frac{1}{4}\right)^x\) demonstrates how, with each increase in \(x\), the output drops, being divided by 4.