Problem 10 - 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. $$f(x)=10^{x}$$

Step-by-Step Solution

Answer

The graph of the function \(f(x) = 10^x\) is an increasing function with a y-intercept at (0,1) and no x-intercept. The x-axis (y=0) is a horizontal asymptote of the function.

Step 1: Determine a range of x values and calculate corresponding y values

Choose a range of x values. Given the graph is exponential, a good range might be from -2 to 2. Then, calculate the corresponding y values using the function \(f(x) = 10^x\). For instance, when x=-2, y will be \(10^{-2} = 0.01\). When x=0, y=1. And when x=2, y=100. These points will help to plot the graph.

Step 2: Plot these points and sketch the graph

After determining the set of points (-2,0.01), (0,1), and (2,100), plot these points on graph paper. Keep in mind that \(10^x\) is an exponential function and it grows rapidly as x increases. Draw the curve to reflect this characteristic of rapid growth as x increases. Connect the points with a smooth curve.

Step 3: Identify the asymptote

A horizontal asymptote is a horizontal line that the graph approaches as x moves towards positive or negative infinity. For this function, as x approaches negative infinity, y tends to 0 but never reaches it since there is no x value that will make the function equal to zero. So, the line y = 0 or the x-axis is a horizontal asymptote of this function.

Step 4: Determine intercepts

The x-intercept(s) of the function are the x value(s) that make the function equal to zero. Since there's no x that can make \(10^x = 0\), there is no x-intercept. The y-intercept is the value of the function when x = 0. For this function, \(y = 10^0 = 1\), so the y-intercept is at point (0,1).

Step 5: Determine if graph is increasing or decreasing

Finally, look at the form of the graph to conclude if it is increasing or decreasing function. Since the graph moves up to the right (the y-values increase as x increases), we can say that \(10^x\) is an increasing function.

Key Concepts

Understanding Asymptotes

In mathematical terms, an asymptote is a line that a graph approaches but never actually touches. In the context of the exponential function \(f(x) = 10^x\), as the value of \(x\) becomes more negative, the function values (or \(y\) values) get closer and closer to zero. However, they never really become zero.

The horizontal line \(y = 0\)—also known as the x-axis—serves as the horizontal asymptote for this function. This is because as \(x\) tends toward negative infinity, the function value \(10^x\) approaches zero. Understanding asymptotes is crucial for predicting the behavior of functions at extreme values of \(x\). How close a function gets to an asymptote gives insight into its long-term behavior.

Intercepts of the Graph

Intercepts are points where the graph of a function crosses the axes. In the exponential function \(f(x) = 10^x\), let's identify these crucial points:

• Y-intercept: This is the point where the graph crosses the y-axis. For \(f(x) = 10^x\), when \(x\) is zero, \(y = 10^0 = 1\). Therefore, the y-intercept is at (0,1).


• X-intercept: This is the point where the graph crosses the x-axis. In this function, there is no x value that makes \(10^x = 0\). Thus, there are no x-intercepts for this exponential function.

Determining the intercepts helps to anchor the graph and provides a clearer picture of its placement relative to the axes.

Exponential Growth

Exponential growth is a process that increases quantity at a consistent rate every fixed period of time. In our function \(f(x) = 10^x\), the base value of 10 means that for every increase in \(x\) by 1 unit, the output \(y\) is multiplied by 10.

The growth is rapid and dramatic. For example, as you move from \(x = 0\) to \(x = 1\), the y-value increases from 1 to 10, and from \(x = 1\) to \(x = 2\), it jumps to 100. This pattern continues, showing how exponential functions can model processes that grow quickly, such as populations or financial investments where compounding interest is involved.

Graphing Exponential Functions

Graphing exponential functions like \(f(x) = 10^x\) is straightforward once you understand the steps. Here’s how you do it:

• Calculate and Plot Points: Choose a range for \(x\), such as from \(-2\) to \(2\), and compute corresponding \(y\) values. Plot these points on the graph. This range, especially for exponential functions, will give a clear picture of the graph's rise on the positive side.


• Draw the Curve: Once the important points are plotted, smoothly connect them. Remember that exponential functions show a rapid increase. Your graph should reflect this steep upward movement as \(x\) goes from negative to positive.


• Recognize the Asymptote: Keep in mind that the line \(y = 0\) is your horizontal asymptote. Ensure your graph approaches this line as \(x\) becomes more negative.

This visualization helps you understand not only the specific function you are dealing with but also the nature of exponential functions in general.