Problem 15 - Algebra and Trigonometry Real Mathematics, Real People

Question

Graph the exponential function by hand. Identify any asymptotes and intercepts and determine whether the graph of the function is increasing or decreasing. $$g(x)=\left(\frac{5}{4}\right)^{-x}$$

Step-by-Step Solution

Answer

The function \( g(x)=(\frac{5}{4})^{-x} \) or equivalently \( (\frac{4}{5})^{x} \) has a y-intercept at (0,1) and no x-intercept. The x-axis (y=0) is the horizontal asymptote. It is a decreasing function.

Step 1: Identify the intercepts

The y-intercept of the function occurs at \( x=0 \). Plugging \( x=0 \) into the function gives \( g(0)=(\frac{4}{5})^{0}=1 \). Hence, the y-intercept is (0,1). This function does not have any x-intercepts as the function never touches the x-axis.

Step 2: Identify the asymptotes

The graph of this function has a horizontal asymptote. As \( x \) approaches \( -\infty \), the function \( (\frac{4}{5})^{x} \) approaches \( \infty \). This implies that the x-axis, or y=0, is a horizontal asymptote.

Step 3: Determine whether the graph is increasing or decreasing

Since the base of the exponent, \( \frac{4}{5} \), is less than 1, the function represents exponential decay. This means it decreases as \( x \) increases.

Key Concepts

Graphing Exponential Functions

Graphing exponential functions, especially by hand, is all about understanding the function's behavior. Here we have the function \( g(x) = \left(\frac{5}{4}\right)^{-x} \). To graph this function, it's helpful to recognize the transformation applied to the base. Notice that \( \left(\frac{5}{4}\right)^{-x} \) is equivalent to \( \left(\frac{4}{5}\right)^{x} \), which indicates an exponential decay.

• Start by plotting the y-intercept, which occurs at \( x=0 \), giving us the point (0,1).
• As \( x \) increases, observe how the function behaves to determine the direction of the curve's decrease.
• Sketch the curve, ensuring it approaches but never crosses the identified asymptote and passes through known points like the y-intercept.

Remember, exponential curves have smooth, gradually increasing or decreasing shapes that never quite touch the x-axis.

Asymptotes

An asymptote is a line that a function approaches but never actually touches. For our function \( g(x) = \left(\frac{5}{4}\right)^{-x} \) or \( g(x) = \left(\frac{4}{5}\right)^{x} \), the horizontal asymptote is the x-axis, or the line \( y=0 \).The horizontal asymptote can be determined by examining the behavior of the function as \( x \) becomes extremely negative or positive.

• For exponential decay, as \( x \to \infty \), the value of \( \left(\frac{4}{5}\right)^{x} \) approaches zero but never actually reaches it.
• This tells us that our function will get closer and closer to the x-axis without touching it, confirming the presence of a horizontal asymptote at \( y = 0 \).

Understanding asymptotes is crucial in graphing as they guide your sketch of the function's long-term behavior.

Intercepts

Intercepts are the points where the graph crosses the axes. For the function \( g(x) = \left(\frac{5}{4}\right)^{-x} \), let's focus on the y-intercept. The y-intercept occurs where \( x = 0 \). Plugging \( x = 0 \) into the function gives:\[g(0) = \left(\frac{4}{5}\right)^{0} = 1\]Hence, the y-intercept is at the point \( (0, 1) \).

• Unlike linear or polynomial functions, this exponential function doesn’t have an x-intercept because it never touches the x-axis. The function approaches infinity or zero but never crosses the x-axis.
• Knowing the intercept is essential for starting your graph, as it provides a definite point the curve will pass through.

Keep these points in mind to accurately depict where the function interacts with the axes.

Exponential Decay

Exponential decay occurs when the value of a function decreases rapidly at first and then more slowly over time. For our function, \( g(x) = \left(\frac{5}{4}\right)^{-x} \) simplifies to \( g(x) = \left(\frac{4}{5}\right)^{x} \). This is a classic example of exponential decay because the base, \( \frac{4}{5} \), is a fraction less than one.This leads to the function value getting smaller as \( x \) gets larger:

• The smaller the base, the faster the rate of decay.
• You can think of it as the graph decreasing as \( x \) increases, shrinking towards the horizontal asymptote.
• Recognizing whether a function is increasing or decreasing is key to forming a complete understanding of its behavior.

By identifying exponential decay, you know the function approaches zero as \( x \to \infty \), characterized by the curve's sharp initial drop tapering off as it extends rightward.