Problem 52 - Introductory and Intermediate Algebra for College Students 4th

Question

Graph \(y=3^{x}\) and \(x=3\) in the same rectangular coordinate system.

Step-by-Step Solution

Answer

The graph of \(y=3^{x}\) is an increasing exponential curve which passes through the point (0, 1) and approaches but never touches the x axis. The graph of \(x=3\) is a vertical line passing through the point (3, 0). When graphed on the same axes, the line \(x=3\) intersects the curve \(y=3^{x}\) at the point (3, 27).

Step 1: Graphing the Exponential Function

Begin with graphing \(y=3^{x}\), the exponential function: a. Draw a horizontal line, the x-axis. Label it as x; b. Draw a vertical line, the y-axis. Label it as y; c. The graph will pass through the point (0, 1) because any number to the power of 0 is 1. This will serve as the y-intercept; d. The graph will be above the x-axis for any \(x > 0\) because 3 raised to any positive power is always more than 0; e. The graph will approach but never touch the x-axis as x gets more and more negative, because any number raised to a very large negative power is essentially zero; f. Draw the curve of the function making sure it passes through the point (0,1) and follows the characteristics detailed in steps c to e.

Step 2: Graphing the Vertical Line

Now, let's graph the equation \(x=3\), the vertical line: a. Draw and extend a vertical line through the point where \(x=3\) on the x-axis; b. This line represents all points in the form (3, y), where y can be any real number.

Step 3: Combining the Graphs

Combine the graphs made in Steps 1 and 2 in the same coordinate plane. Take note where they intersect if they do.

Key Concepts

Coordinate System

Understanding the coordinate system is crucial for graphing functions accurately. A coordinate system, often referred to as a Cartesian coordinate system, is a two-dimensional plane featuring two perpendicular lines that intersect at the origin. These lines are named the x-axis (horizontal) and the y-axis (vertical).

Each point in this system is represented by a pair of numerical coordinates, which are the distances to the point from the x-axis and the y-axis. The intersection, or the origin, has coordinates (0, 0). In the case of graphing the exponential function and the vertical line, you would plot points on this system to reflect their relationship and the independent variable's influence on the dependent variable.

Exponential Function Properties

Exponential functions, like the one in the equation \(y=3^{x}\), have several unique properties. Firstly, the base, 3 in this case, must be a positive number other than 1. One characteristic of this type of function is that it will always pass through the point (0, 1), as any number to the power of zero equals one. This is why \(3^0 = 1\) serves as the y-intercept for the graph.

An exponential function will increase rapidly as the value of x becomes larger. Consequently, it will get closer and closer to the x-axis, but never touch it, as x becomes large in the negative direction. This illustrates an important concept known as 'asymptotic behavior' where the curve approaches an axis—but never reaches it—illustrating that the outcome is approaching zero but doesn't equal zero when \(x\) is highly negative.

Vertical Line Graph

Testing for Functionality with a Vertical Line

Graphing the vertical line \(x=3\) is relatively straightforward. This line runs parallel to the y-axis and crosses the x-axis at the point (3, 0). The vertical line represents all the points where the x-coordinate is 3 and the y-coordinate can be any value.

What's significant about the vertical line in the context of functions is that if any vertical line crosses a graph more than once, the graph cannot represent a function. This is known as the 'Vertical Line Test'. A function, by definition, can only have one output value for any given input value. When graphing, one uses the intersection of the vertical line with another graph to find specific points of interest, such as solutions to the system of equations or to identify the limits of a function.