Problem 14 - 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. $$g(x)=\left(\frac{3}{2}\right)^{x}$$

Step-by-Step Solution

Answer

The graph of \(g(x)=(\frac{3}{2})^{x}\) has a horizontal asymptote at y=0. It intercepts the y-axis at the point (0,1) and doesn't intercept the x-axis. The function is increasing because the base of the exponent (\frac{3}{2}) is greater than 1.

Step 1: Identify the Asymptotes

Start by identifying any asymptotes of the function. For this function, the horizontal asymptote can be found by determining the value of the function as x approaches negative infinity. Since any positive number raised to a power of negative infinity approaches 0, the horizontal asymptote is y=0.

Step 2: Find the Intercepts

Next, find the y-intercept of the function by replacing x with 0, because the y-intercept is where the function crosses the y-axis (i.e., when x=0). Plugging 0 for x in the function \(g(x)=(\frac{3}{2})^{x}\) gives \(g(0)=(\frac{3}{2})^{0}\), which equals 1. So, the function intercepts the y-axis at the point (0,1). The x-intercept of an exponential function with a positive base doesn't exist as the function will never touch the x-axis.

Step 3: Determine the Direction of the Graph

Determine if the function is increasing or decreasing. Since the base of the function (\frac{3}{2}) is greater than 1, this implies that as x increases, \(g(x)\) also increases. Therefore, the function is increasing.

Step 4: Sketch the Graph

Now, using the information gathered in the previous steps, sketch the graph of the function. Start by drawing the horizontal asymptote at y=0. Then plot the y-intercept at the point (0,1). Finally, because the function is increasing, draw a curve starting from the asymptote and going through the y-intercept, directed to the upper right of the graph.

Key Concepts

Exponential Function Intercepts

When studying exponential functions, understanding the intercepts is crucial for graphing them accurately. An intercept is a point where the graph of the function crosses either the x-axis (x-intercept) or the y-axis (y-intercept).

For the function provided in the exercise, \( g(x) = \left(\frac{3}{2}\right)^x \), you can find the y-intercept by setting \( x = 0 \). This is because the y-axis corresponds to the vertical line where \( x = 0 \). In this case, \( g(0) = \left(\frac{3}{2}\right)^0 \), which simplifies to \( g(0) = 1 \). Thus, the y-intercept is at the point (0, 1). This tells us where the graph crosses the y-axis.

On the contrary, the x-intercept, where the graph would cross the x-axis, does not exist for this function. Exponential functions with a positive base greater than 1, like \( \frac{3}{2} \), never reach the value of 0 on the y-axis; therefore, they do not intersect the x-axis at any point. Remember, the x-intercept is the point where \( y = 0 \), and as this function approaches infinity, \( y \) gets indefinitely larger, without ever reaching zero.

Horizontal Asymptotes

A horizontal asymptote is a horizontal line that a graph approaches as \( x \) goes to infinity or negative infinity but never actually touches. For the function \( g(x) = \left(\frac{3}{2}\right)^x \), the horizontal asymptote can be identified by evaluating the limit of \( g(x) \) as \( x \) approaches negative infinity.

Since the base \( \frac{3}{2} \) is a positive number, as \( x \) approaches negative infinity, the value of \( g(x) \) approaches 0. This gives us our horizontal asymptote, located at \( y = 0 \). In other words, no matter how far you extend the graph in either direction along the x-axis, it will get closer and closer to the line \( y = 0 \) but not cross it. This unseen boundary affects the shape and behavior of the graph, giving us vital information about the function's long-term behavior.

Increasing and Decreasing Functions

When dealing with a function like \( g(x) = \left(\frac{3}{2}\right)^x \), understanding whether it's increasing or decreasing is fundamental for determining its growth trend. For exponential functions, this aspect depends entirely on the base of the exponential term.

An exponential function with a base greater than 1 is always increasing. This means as \( x \) gets larger, \( g(x) \) also gets larger. Conversely, if the base was between 0 and 1, the function would decrease as \( x \) increases. The function in our exercise, with a base of \( \frac{3}{2} \), is an increasing function because \( \frac{3}{2} > 1 \). This is important to know when graphing, as it indicates that the curve of the graph will rise as we move from left to right across the graph.

This aspect alongside the horizontal asymptote and y-intercept creates the distinctive 'J-shaped' curve of an increasing exponential function. Recognizing these characteristics helps in sketching the graph by hand and predicting the function's growth over time.