Problem 19 - Introductory and Intermediate Algebra for College Students 4th

Question

Graph each function by making a table of coordinates. If applicable, use a graphing unility to confirm your hand-drawn graph. $$g(x)=\left(\frac{3}{2}\right)^{x}$$

Step-by-Step Solution

Answer

The graph of the function \(g(x) = (\frac{3}{2})^{x}\) starts relatively flat to the left, passes through the point (0,1), and then climbs steeply to the right. This indicates that it is an exponential growth function.

Step 1: Understand the function

The function given is \(g(x) = (\frac{3}{2})^{x}\), which is an exponential function. In this function, x is the input and y is the output. The base of the exponent is \(\frac{3}{2}\). Our task is to substitute different values of x into the function, calculate the corresponding y values, and then plot those points to graph the function.

Step 2: Construct a table of coordinates

Let's construct a table of coordinates by choosing different x values and computing the corresponding y values. For example:\n - For \(x=-2\), \(g(-2)= (\frac{3}{2})^{-2}= (\frac{2}{3})^{2}= \frac{4}{9}\), so the coordinate is \(-2,\frac{4}{9}\).\n - For \(x=-1\), \(g(-1)= (\frac{3}{2})^{-1}= \frac{2}{3}\), so the coordinate is \(-1,\frac{2}{3}\).\n - For \(x=0\), \(g(0)= (\frac{3}{2})^{0}= 1\), so the coordinate is \(0,1\).\n - For \(x=1\), \(g(1)= (\frac{3}{2})^{1}= \frac{3}{2}\), so the coordinate is \(1,\frac{3}{2}\).\n - For \(x=2\), \(g(2)= (\frac{3}{2})^{2}= \frac{9}{4}\), so the coordinate is \(2,\frac{9}{4}\).

Step 3: Plot the coordinates on a graph

Now that we have multiple points, let's plot them on a graph. Next, draw a smooth curve that goes through all points. The shape of the curve should represent an exponential growth since the base of the exponent (\(\frac{3}{2}\)) is greater than 1.

Step 4: Confirm the graph using a graphing utility

To ensure our graph is correct, let's confirm it using a graphing utility, such as a graphing calculator or an online graphing tool. The general shape of the function should be the same as your hand-drawn graph, starting relatively flat to the left, passing through the point (0,1), and then climbing steeply to the right.

Key Concepts

Graphing

When graphing exponential functions like \( g(x) = \left(\frac{3}{2}\right)^{x} \), one of the first steps is to establish a set of points by making a table of coordinates. This allows us to visualize how the function behaves.
Start by choosing a range of values for x, both negative and positive, to see how the y-values (the results of \( g(x) \)) change:

• For \( x = -2 \), the y-value is \( \frac{4}{9} \)
• For \( x = -1 \), the y-value is \( \frac{2}{3} \)
• For \( x = 0 \), the y-value is 1
• For \( x = 1 \), the y-value is \( \frac{3}{2} \)
• For \( x = 2 \), the y-value is \( \frac{9}{4} \)

Once you have these coordinates, you can plot them on a graph. Connect the dots smoothly, as exponential graphs have curves that are not linear. Exponential functions typically rise rapidly or decline sharply depending on the base. Since here the base \( \frac{3}{2} \) is greater than one, the graph shows exponential growth.

Coordinate System

The coordinate system is essential for graphing functions. It consists of two axes: the horizontal (x-axis) and the vertical (y-axis). To graph a function, like \( g(x) = \left(\frac{3}{2}\right)^{x} \), you need to understand this system to correctly place your points.

• **X-axis**: This is the horizontal line where you will place your x-values. Positive values extend to the right, and negative values extend to the left.
• **Y-axis**: This is the vertical line, and you will place your computed y-values here. Positive values go up, and negative values go down.

To plot a point, find your x-value on the x-axis, and your y-value on the y-axis. Where these intersect is where you place your point (x, y). With the function \( g(x) = \left(\frac{3}{2}\right)^{x} \), use the table of coordinates you made to find points like (-2, \( \frac{4}{9} \)) or (1, \( \frac{3}{2} \)). By plotting all these points, the graph slowly takes shape into the curve of the exponential function.

Exponential Growth

Exponential growth is a pattern where quantities increase rapidly over time, often doubling or more, rather than increasing steadily. When graphing an exponential growth function like \( g(x) = \left(\frac{3}{2}\right)^{x} \), this means the function's y-values grow quickly as the x-values increase.
In mathematical terms, when the base of the exponential (in this case, \( \frac{3}{2} \)) is greater than one, the function models exponential growth. This causes the graph to:

• Start off slowly for negative x-values, appearing almost flat.
• Cross the y-axis at 1 when x is zero, because \( \left(\frac{3}{2}\right)^{0} = 1 \).
• Rise rapidly as x becomes positive, steepening sharply.

Such functions are used in real-world scenarios, such as modeling population growth, where numbers can expand rapidly. Understanding how to create and interpret these graphs is very helpful for seeing these growth trends and making predictions.