Problem 50 - Algebra and Trigonometry

Question

graph functions f and g in the same rectangular coordinate system. Graph and give equations of all asymptotes. If applicable, use a graphing utility to confirm your hand-drawn graphs. $$ f(x)=3^{x} \text { and } g(x)=3 \cdot 3^{x} $$

Step-by-Step Solution

Answer

The graphs of the functions f(x) = 3^{x} and g(x) = 3 * 3^{x} are exponential growth curves with y-intercepts at (0,1) and (0,3) respectively. Both of them have the x-axis (y = 0) as their asymptote.

Step 1: Identify The General Form

The functions given are exponential in nature. An exponential function has a general form a * b^{x}, where a is the y-intercept, b is the base and x is the independent variable. Here for f(x), a = 1 and b = 3. For g(x), a = 3 and b = 3.

Step 2: Plot The Function f(x) = 3^{x}

For f(x) = 3^{x}, it cuts the y-axis at y = 1 (as for x=0, f(x)=1). It is upwards sloping indicating that as x increases, y too increases. Also, for x\(<\)0, as x becomes more negative, y (3^{x}) approaches zero, thus y = 0 is the asymptote.

Step 3: Plot The Function g(x) = 3 * 3^{x}

For g(x) = 3 * 3^{x}, it cuts the y-axis at y = 3 (as for x=0, g(x)=3). Similarly, it's a rapidly increasing function. And like f(x), for x\(<\)0, the function value of g(x) approaches zero, thus y = 0 is the asymptote.

Step 4: Identify The Asymptotes

As outlined in previous steps, for both the functions f(x) = 3^{x} and g(x) = 3 * 3^{x}, the x-axis (y = 0) is the asymptote. This is because as x becomes more and more negative, the value of both functions approaches zero.

Step 5: Confirm With Graphing Utility

Use a graphical utility to plot these functions and observe the exact graph behavior. It should align with our hand-drawn understanding.

Key Concepts

Asymptotes

Exponential functions often involve a fundamental concept known as asymptotes. An asymptote is a line that a graph approaches but never quite reaches. In the case of functions like \(f(x) = 3^{x}\) and \(g(x) = 3 \cdot 3^{x}\), you will find horizontal asymptotes.For these particular functions:

• The horizontal asymptote is the x-axis, or \(y = 0\).

This is because as \(x\) decreases into negative values, the value of \(3^{x}\) also decreases, approaching zero but never actually reaching it. Even when \(x\) is very large in the negative direction, \(3^{x}\) gets very close to zero but remains positive. This behavior is characteristic of exponential functions that do not have any vertical asymptotes. Understanding this allows you to see that although the x-axis is a boundary the graph never crosses, it doesn't limit the upward growth in the positive x-direction.

Coordinate System

In mathematics, a coordinate system is essential for visually representing functions. The rectangular coordinate system, also known as the Cartesian coordinate system, is widely used for plotting graphs. It consists of two perpendicular lines called axes:

• The horizontal axis (x-axis)
• The vertical axis (y-axis)

Each point on this plane is defined by a pair of numbers \(x, y\). In the context of the given functions \(f(x) = 3^{x}\) and \(g(x) = 3 \cdot 3^{x}\), plotting involves identifying points where both functions intersect these axes.For \(f(x) = 3^{x}\):- It intersects the y-axis at \((0, 1)\) because \(f(0) = 1\).For \(g(x) = 3 \cdot 3^{x}\):- It intersects the y-axis at \((0, 3)\) because \(g(0) = 3\).Understanding this system helps us not only in plotting points but also in understanding the behavior of these exponential functions in different quadrants of the plane.

Graphing Utility

Graphing utilities are powerful tools that help in visualizing mathematical functions and their properties. A graphing utility can be either an online application or a physical calculator equipped with graphing capabilities. These tools enhance the understanding of functions beyond manual plotting. When using a graphing utility for exponential functions like \(f(x) = 3^x\) and \(g(x) = 3 \cdot 3^x\):

• Input the equations as they are.
• The utility will automatically plot the functions.
• It visually confirms the horizontal asymptote at \(y = 0\).

These utilities are especially helpful when graph details are complex, allowing us to see potential symmetries, intercepts, and variations on a large scale. With a graphing utility, you can also easily test your hand-drawn graphs' accuracy, providing a bridge between conceptual understanding and practical application. They are particularly invaluable in verifying results and developing a stronger grasp of mathematical concepts.