Problem 33 - Introductory and Intermediate Algebra for College Students 4th

Question

Graph functions \(f\) and \(g\) in the same rectangular coordinate system. Select integers from \(-2\) to 2 , inclusive, for \(x\). Then describe how the graph of g is related to the graph of \(f .\) If applicable, use a graphing utility to confirm your hand-drawn graphs. $$f(x)=3^{x} \text { and } g(x)=-3^{x}$$

Step-by-Step Solution

Answer

The graph of \(g(x)=-3^{x}\) is a mirror reflection of the graph of \(f(x)=3^{x}\) along the x-axis.

Step 1: Prepare the values

Identify the integer values of \(x\) from -2 to 2 inclusive. These are -2, -1, 0, 1, and 2.

Step 2: Compute \(f(x)\) and \(g(x)\)

Compute the values of the functions \(f(x)=3^{x}\) and \(g(x)=-3^{x}\) for each \(x\) value. For \(f(x)\), the resulting values of \(y\) are \(\frac{1}{9}, \frac{1}{3}, 1, 3, 9\), respectively. For \(g(x)\), the values of \(y\) are \(-\frac{1}{9}, -\frac{1}{3}, -1, -3, -9\), respectively.

Step 3: Graph the functions

Graph both functions in the same coordinate system. Remember \(f(x)\) will be in the first and second quadrant (as \(y\) is positive), and \(g(x)\) will be in the third and fourth quadrant (as \(y\) is negative).

Step 4: Analyze the graphs

Observe both graphs. \(f(x)\) and \(g(x)\) have similar shapes, but \(g(x)\) is a reflection of \(f(x)\) in the x-axis due to the negative sign.

Step 5: Confirm with a graphing utility

If possible, use a graphing utility to confirm the above steps. It should show the same results – \(f(x)\) having a growth in the positive direction and \(g(x)\) having a mirrored growth in the negative direction.

Key Concepts

Coordinate System

Understanding the rectangular coordinate system, also known as the Cartesian coordinate system, is fundamental to graphing any function, including exponential functions. The coordinate system consists of two perpendicular number lines: the horizontal axis (x-axis) and the vertical axis (y-axis). Their intersection point is called the origin, with coordinates \( (0, 0) \). Each point in this system is defined by an ordered pair of numbers \( (x, y) \), representing its horizontal and vertical displacements from the origin.

Graphing an exponential function like \(f(x) = 3^x\) involves plotting points obtained by substituting integer values of \(x\) within the given range. In the given exercise, the range is from \(x = -2\) to \(x = 2\). By calculating \(f(x)\) for each \(x\), you get corresponding \(y\) values. These ordered pairs \( (x, f(x)) \) form the points you plot on the grid. With all points marked, you draw a curve through them to visualize the exponential growth.

Reflection of Functions

Reflection of functions is a transformation that creates a mirror image of a given function across a certain line, such as the x-axis or y-axis. For the function \(g(x) = -3^x\), the negative sign indicates that \(g(x)\) is a reflection of \(f(x) = 3^x\) across the x-axis. While \(f(x)\) graphs into the first and second quadrants, \(g(x)\), being its reflection, maps into the third and fourth quadrants.

This reflection alters the graph's vertical coordinate \(y\), but leaves the \(x\)-coordinate unchanged. As a result, for every point \( (x, y) \) on the graph of \(f(x)\), there is a corresponding point \( (x, -y) \) on the graph of \(g(x)\). This is a crucial concept when studying the behaviors of different types of transformations in functions.

Exponential Growth and Decay

Exponential growth and decay are fundamental concepts in mathematics embodying how quantities increase or decrease rapidly. In exponential growth, the rate of change of a quantity is directly proportional to the current amount, leading to the increasing steepness of the graph, represented by the function \(f(x) = 3^x\) in our exercise. As \(x\) increases, the \(y\) values grow exponentially.

Conversely, exponential decay describes a quantity decreasing rapidly, which would be represented by a graph getting closer to the x-axis as \(x\) increases. However, in this exercise, the function \(g(x) = -3^x\) does not represent decay but rather growth in the negative \(y\)-direction, due to the reflection over the x-axis. It's important to distinguish between negative growth, which is still a form of exponential growth, and decay, which is demonstrated by functions with fractions as bases, such as \( (1/3)^x\).