Problem 495 - Precalculus

Question

Graph the function \(f(x)=4\left(\frac{1}{8}\right)^{x}\) and its reflection about the \(y\) -axis on the same axes, and give the \(y\) -intercept.

Step-by-Step Solution

Answer

The y-intercept is (0, 4).

Step 1: Identify the Function Type

The given function \( f(x) = 4 \left(\frac{1}{8}\right)^{x} \) is an exponential function. It has the form \( a(b)^x \), where \( a = 4 \) and \( b = \frac{1}{8} \). Since \( 0 < b < 1 \), the function is a decreasing exponential function.

Step 2: Determine the y-intercept of f(x)

To find the \( y \)-intercept, set \( x = 0 \) in the function. \[ f(0) = 4 \left(\frac{1}{8}\right)^{0} = 4 \times 1 = 4. \] So, the \( y \)-intercept is \( (0, 4) \).

Step 3: Graph f(x)

The graph of \( f(x) = 4 \left(\frac{1}{8}\right)^{x} \) is an exponentially decreasing curve. It passes through the point \( (0, 4) \) and approaches the \( x \)-axis as \( x \) increases.

Step 4: Find the Reflection f(-x)

Reflect the graph of \( f(x) \) across the \( y \)-axis by replacing \( x \) with \( -x \) in the function. The reflection is \( f(-x) = 4 \left(\frac{1}{8}\right)^{-x} = 4 \cdot 8^x \). This creates a new function that is exponentially increasing.

Step 5: Graph the Reflection f(-x)

The graph of \( f(-x) = 4 \cdot 8^x \) is an exponentially increasing curve. It passes through the same \( y \)-intercept \( (0, 4) \) as \( f(x) \) and increases rapidly as \( x \) increases.

Key Concepts

Exponential Functions

Exponential functions are quite fascinating, as they represent situations where a quantity grows or decays at a rate proportional to its current value. The general form of an exponential function is given by \(f(x) = a(b)^x\). In this formula:

• \(a\) is a constant, indicating the initial value or scale factor.
• \(b\) is the base of the exponential, determining the growth or decay factor.
• \(x\) is the exponent, and it varies to show different values of the function.

Consider the given function, \(f(x) = 4 \left(\frac{1}{8}\right)^x\), where \(a = 4\) and \(b = \frac{1}{8}\). The function starts at the value 4 when \(x = 0\). The base \(b < 1\) indicates a process of exponential decay. By understanding these components, you can easily recognize and handle exponential functions in various contexts.

Decreasing Function

The function \(f(x) = 4 \left(\frac{1}{8}\right)^x\) is an example of a decreasing exponential function. This means that as \(x\) increases, the value of \(f(x)\) decreases. Here's why:

• In our function, the base \(b\) is \(\frac{1}{8}\), which is less than 1.
• Since \(0 < b < 1\), the function decreases as \(x\) increases.
• Therefore, the values of \(f(x)\) approach zero, but never touch the x-axis.

This behavior is useful in modeling real-world phenomena such as reducing populations, depleting resources, or cooling processes. Knowing that \(f(x)\) decreases can help you predict how quickly the function's values decline and understand its long-term behavior.

Reflection across Y-Axis

Reflecting a function across the y-axis provides a mirror image of the function's graph. To perform this reflection for the function \(f(x) = 4 \left(\frac{1}{8}\right)^x\), replace \(x\) with \(-x\). This process results in a new function, \(f(-x) = 4 \cdot 8^x\).

• This transformation involves switching the base from \(\frac{1}{8}\) to 8, and the behavior changes from decreasing to increasing.
• Reflecting across the y-axis affects the direction of the exponential behavior; what was a decay is now a rapid increase.
• This new function still passes through the same y-intercept \((0, 4)\), but grows exponentially for positive values of \(x\).

Reflection across the y-axis is a key transformation that helps visualize how a function's behavior changes direction.

Y-Intercept

The y-intercept of a function is a critical point on its graph - the location where the graph crosses the y-axis. For exponential functions, finding the y-intercept is straightforward. We calculate it by setting \(x = 0\) and evaluating the function.

• For \(f(x) = 4 \left(\frac{1}{8}\right)^x\), when \(x = 0\), \(f(0) = 4 \cdot 1 = 4\).
• This result tells us that the y-intercept is \((0, 4)\).
• Even after a reflection, as in \(f(-x) = 4 \cdot 8^x\), the y-intercept remains \((0, 4)\), unchanged by this particular transformation.

By understanding the y-intercept, we gain a pivotal reference for comparing functions and analyzing their starting points on a graph.