91Ó°ÊÓ

Exponential growth occurs when the base of an exponential function, represented as \(a\) in the form \(y = C \times a^{x}\), is greater than 1. This means that for each unit increase in \(x\), the value of \(y\) multiplies by a constant factor. For the given functions in the exercise: \(y = 2^{x}\), \(y = 5^{x}\), and \(y = 10^{x}\), all exhibit exponential growth.
Here's a quick look at what happens:
* When the base is 2, \(y\) doubles.
* When the base is 5, \(y\) quintuples.
* When the base is 10, \(y\) decuples.

This rapid increase is what makes exponential functions especially powerful and sometimes surprising.
Always remember, exponential growth means quicker increases as \(x\) becomes larger.

A horizontal asymptote is a line that a graph approaches but never touches as \(x\) approaches infinity or negative infinity. For exponential functions, the horizontal asymptote can give us insight into the long-term behavior of the graph.

In our given functions: \(y = 2^{x}\), \(y = 5^{x}\), and \(y = 10^{x}\), the horizontal asymptote is \(y = 0\). This means as \(x\) approaches negative infinity (\text{-∞}), the value of \(y\) will get closer and closer to zero without ever actually reaching zero.
This happens because when you raise a positive number greater than 1 to an increasingly negative power, the result gets smaller and smaller, heading toward zero.
Therefore, no matter how far along the negative \(x\)-axis you go, the curve will continue to approach the \(x\)-axis but never touch it.

Graphing exponential functions involves plotting points to see the shape of the curve. Let’s use our three functions as examples:
1. \(y = 2^{x}\) starts at (0, 1) and doubles as x increases, forming a curve that gets steeper and steeper.
2. \(y = 5^{x}\) starts also at (0, 1) but increases much faster than \(2^{x}\) due to its larger base.
3. \(y = 10^{x}\) starts at (0, 1) and grows even more rapidly.

Here’s how you can graph these functions:
* Start by creating a set of points based on specific values for \(x\). For instance: - For \(y = 2^{x}\) at \(x = -1\), \(y = 0.5\); \(x = 0\), \(y = 1\); \(x = 1\), \(y = 2\); \(x = 2\), \(y = 4\).
* Plot the points on graph paper or using graphing software.
* Connect the dots smoothly to illustrate the exponential curve.

By graphing all three functions on the same grid, we can confirm that \(y = 10^{x}\) will be the top curve, \(y = 5^{x}\) in the middle, and \(y = 2^{x}\) on the bottom.

Creating a table of values is essential for understanding and plotting exponential functions. By choosing a range of \(x\) values and calculating the corresponding \(y\) values, you can easily map out the curve.

Let’s look at tables of values for our functions:
* For \(y = 2^{x}\):
- \(x = -1\), \(y = 0.5\)
- \(x = 0\), \(y = 1\)
- \(x = 1\), \(y = 2\)
- \(x = 2\), \(y = 4\)
* For \(y = 5^{x}\):
- \(x = -1\), \(y = 0.2\)
- \(x = 0\), \(y = 1\)
- \(x = 1\), \(y = 5\)
- \(x = 2\), \(y = 25\)
* For \(y = 10^{x}\):
- \(x = -1\), \(y = 0.1\)
- \(x = 0\), \(y = 1\)
- \(x = 1\), \(y = 10\)
- \(x = 2\), \(y = 100\)

A table helps visualize how quickly the \(y\) values increase and serves as a great aid in plotting the graph for exponential functions.