Problem 14 - Algebra and Trigonometry

Question

11–14 ? Graph both functions on one set of axes. $$ f(x)=\left(\frac{2}{3}\right)^{x} \quad \text { and } \quad g(x)=\left(\frac{4}{3}\right)^{x} $$

Step-by-Step Solution

Answer

Graphs: \( f(x) \) decreases, \( g(x) \) increases; intersect at \( (0,1) \).

Step 1: Identify the Type of Functions

The functions given are exponential functions. They have the form \( f(x) = a^x \), where \( a \) is a constant. In our case, \( f(x) = \left(\frac{2}{3}\right)^x \) and \( g(x) = \left(\frac{4}{3}\right)^x \).

Step 2: Determine the Behavior of the Graphs

For \( f(x) = \left(\frac{2}{3}\right)^x \), since \( \frac{2}{3} < 1 \), the graph will be a decreasing exponential function that approaches zero as \( x \) increases. For \( g(x) = \left(\frac{4}{3}\right)^x \), since \( \frac{4}{3} > 1 \), the graph will be an increasing exponential function that grows larger as \( x \) increases.

Step 3: Plot Key Points for Each Graph

For \( f(x) \): Calculate values at key points: \( f(0) = 1 \), \( f(1) = \frac{2}{3} \), and \( f(-1) = \frac{3}{2} \), showing that it decreases. For \( g(x) \): Calculate similarly: \( g(0) = 1 \), \( g(1) = \frac{4}{3} \), and \( g(-1) = \frac{3}{4} \), showing that it increases.

Step 4: Sketch the Graphs on the Axes

Draw a set of axes with \( x \) from \(-2\) to \(2\) and \( y \) from \(0\) to \(2\). Plot the points for \( f(x) \) and connect them to show a downward curve. Plot the points for \( g(x) \) and connect them to show an upward curve. Ensure both curves intersect at \( (0, 1) \).

Step 5: Label and Analyze the Graphs

Label each curve with its function \( f(x) \) or \( g(x) \) for clarity. Analyze that \( f(x) \) intersects the y-axis at \( (0, 1) \) and approaches zero as \( x \) increases, while \( g(x) \) also intersects at \( (0, 1) \) but increases.

Key Concepts

Exponential Growth

Exponential growth occurs when the rate of change of a function is proportional to its current value. The function multiplies by a constant factor greater than one as we progress along the x-axis. In other words, each step in the x-direction results in multiplying the initial amount by the base of the function.
This is seen in the function \( g(x) = \left( \frac{4}{3} \right)^x \). Since the base \( \frac{4}{3} \) is greater than one, the output values of the function increase rapidly as x increases.
Consider these points:

• \( g(0) = 1 \) because any non-zero number raised to the power of 0 is 1.
• \( g(1) = \frac{4}{3} \) as it directly equals the base.
• \( g(-1) = \frac{3}{4} \) illustrates growth in reverse direction as values reduce when x is negative.

The curve of \( g(x) \) starts from \( (0, 1) \) and stretches upwards with increasing steepness, revealing exponential growth.

Exponential Decay

Exponential decay, on the other hand, describes a process where the rate of change is also proportional to the current value, but in this case, the constant factor is less than one. This leads to a decrease in the function's value as x increases.
In the function \( f(x) = \left( \frac{2}{3} \right)^x \), since \( \frac{2}{3} \) is less than one, the output values decrease as x increases.
To understand this better, observe the following:

• \( f(0) = 1 \), just like any number to the power of zero.
• \( f(1) = \frac{2}{3} \), showing the start of the decay.
• \( f(-1) = \frac{3}{2} \), illustrating an increase in backward direction.

The graph of \( f(x) \) starts from \( (0, 1) \) and curves downward, flattening out as x grows larger. This demonstrates exponential decay.

Graphing Functions

Graphing exponential functions visually showcases their behavior over different intervals. For both exponential growth and decay, key points help determine the curve's trajectory. When graphing, it's essential to:

• Set up axes with an appropriate range for x and y values.
• Plot calculated key points accurately to outline the trend.
• Understand that the y-intercept always occurs at the point where x equals zero.

Place \( f(x) \) and \( g(x) \) on the same axes to see how they differ. Begin by marking key points such as \( (0,1) \) which they share, and extend their paths according to their increasing or decreasing nature. The intersecting nature at \( (0,1) \) helps to visualize both exponential growth and decay in context.

Function Behavior

To understand the behavior of exponential functions, it's crucial to analyze the effects of their base value. The behavior greatly depends on whether this base is greater or less than 1.
Here are some insights about function behavior:

• For \( f(x) = \left( \frac{2}{3} \right)^x \), the function approaches zero for large positive x, illustrating decay.
• In \( g(x) = \left( \frac{4}{3} \right)^x \), values increase without bound for positive x, indicating growth.
• Observing special points like \( (0,1) \) where both graphs touch gives insight into the starting point of both behaviors.

Considering the entire context, exponential functions show predictable patterns: exponential growth leads to ever-increasing values, while exponential decay results in diminishing returns as x extends. Understanding these patterns in terms of function behavior provides a strong foundation for graph analysis and the prediction of future values.