Problem 79 - Precalculus

Question

For the following exercises, graph each set of functions on the same axes. $$f(x)=3\left(\frac{1}{4}\right)^{x}, g(x)=3(2)^{x}, \text { and } h(x)=3(4)^{x}$$

Step-by-Step Solution

Answer

Graph each function using key points to observe different growth and decay behaviors.

Step 1: Understand the Functions

We have three exponential functions: \( f(x)=3\left(\frac{1}{4}\right)^{x} \), \( g(x)=3(2)^{x} \), and \( h(x)=3(4)^{x} \). Each has a different base inside the exponential.

Step 2: Identify the Key Points

For each function, calculate a few key points. Usually, these points include when \( x = -1, 0, 1, 2 \). This will help you understand the behavior of each function.

Step 3: Calculate Key Points for \( f(x) \)

- For \( x = -1 \), \( f(x) = 3\left(\frac{1}{4}\right)^{-1} = 3 imes 4 = 12 \).- For \( x = 0 \), \( f(x) = 3 \).- For \( x = 1 \), \( f(x) = 3\left(\frac{1}{4}\right) = \frac{3}{4} \).- For \( x = 2 \), \( f(x) = 3\left(\frac{1}{16}\right) = \frac{3}{16} \).

Step 4: Calculate Key Points for \( g(x) \)

- For \( x = -1 \), \( g(x) = 3(2)^{-1} = 3 \times \frac{1}{2} = \frac{3}{2} \).- For \( x = 0 \), \( g(x) = 3 \).- For \( x = 1 \), \( g(x) = 3 \times 2 = 6 \).- For \( x = 2 \), \( g(x) = 3(4) = 12 \).

Step 5: Calculate Key Points for \( h(x) \)

- For \( x = -1 \), \( h(x) = 3(4)^{-1} = 3 \times \frac{1}{4} = \frac{3}{4} \).- For \( x = 0 \), \( h(x) = 3 \).- For \( x = 1 \), \( h(x) = 3(4) = 12 \).- For \( x = 2 \), \( h(x) = 3(16) = 48 \).

Step 6: Graph the Functions

Plot each function on the same set of axes. Use the points calculated in Steps 3, 4, and 5.- \( f(x) \) decreases sharply as \( x \) increases.- \( g(x) \) increases moderately as \( x \) increases.- \( h(x) \) increases sharply, more steeply than \( g(x) \), as \( x \) increases.

Step 7: Analyze the Graphs

Observe that \( f(x) \) shows exponential decay, while \( g(x) \) and \( h(x) \) show exponential growth. \( g(x) \) grows at a moderate rate, whereas \( h(x) \) grows very rapidly.

Key Concepts

Graphing Exponential Functions

When it comes to graphing exponential functions, we are looking at how changes in the base of the exponent affect the graph's shape. The foundational form of an exponential function is usually expressed as \(f(x) = a \cdot b^{x}\), where \(a\) is a constant, and \(b\) is the base. In this exercise, we have three functions, each with the same multiplier \(a = 3\), but different bases: \(f(x)=3\left(\frac{1}{4}\right)^{x}\), \(g(x)=3(2)^{x}\), and \(h(x)=3(4)^{x}\).

To effectively graph these functions, you'll want to calculate key points (like when \(x = -1, 0, 1, 2\)) and plot them on the same set of axes to visualize their behavior. It's also important to note that at \(x = 0\), all three functions have a value of 3, making 3 the common y-intercept.

By graphing, you can see the contrast in how these functions behave as \(x\) increases. Noticing this visually is crucial, as the steepness or flatness of the curve tells us a lot about the function's nature.

Exponential Growth

Exponential growth is a key behavior in functions where the base of the exponential is greater than 1. In our exercise, both \(g(x)=3(2)^{x}\) and \(h(x)=3(4)^{x}\) showcase exponential growth. This means that as \(x\) increases, the value of the function grows very rapidly.

To understand this better, compare the graphs.

• For \(g(x)\), the growth is moderate. Observing the function points, at \(x=2\), \(g(x)\) is 12.
• For \(h(x)\), the growth is faster, and at the same \(x=2\), \(h(x)\) has already reached 48.

The speed of the growth depends heavily on the base. The larger the base, the steeper the growth. This is why \(h(x)\)'s graph is much steeper than \(g(x)\)'s.

Exponential Decay

Exponential decay occurs in functions where the base is between 0 and 1. In the function \(f(x)=3\left(\frac{1}{4}\right)^{x}\), exponential decay is evident. As \(x\) increases, the value of \(f(x)\) actually decreases.

This decay is demonstrated by the key points: at \(x=-1\), \(f(x)\) is 12, but by \(x=2\), it has plummeted to \(\frac{3}{16}\). Functions like \(f(x)\) shrink quickly because each increase in \(x\) results in multiplying by a fraction less than 1. This reduction contrasts starkly with the growth observed in \(g(x)\) and \(h(x)\).

Understanding decay is valuable in fields such as chemistry and physics, where processes like radioactive decay can model similar mathematical behavior.

Function Transformation

In terms of transformations, exponential functions can undergo shifts, stretches, or compressions along the axes. In this context, the functions remain in their basic form \(f(x) = a \cdot b^{x}\), allowing us to focus on the influence of their bases.

The transformations might involve vertical shifts (affected by the constant \(a\)), but importantly in the given functions, \(a\) remains as 3 for all, ensuring no vertical shift. The real transformation lies in the change of the base, affecting the curve steepness and whether the function represents growth or decay.

• A smaller base than 1 results in decay, transforming the graph to descend as \(x\) increases (seen in \(f(x)\)).
• Larger bases lead to growth, where the graph ascends steeply (as seen in \(g(x)\) and even more so in \(h(x)\)).

Visualizing these transformations through graphing offers valuable insights into not just mathematics, but also real-world phenomena that follow exponential patterns.