Identifying Characteristics of the Exponential Function y = bx (b > 1) The domain of an exponential function is . The range of an exponential function is . On a coordinate plane, the graph of y = 2 Superscript x is shown. The curve approaches the x-axis in quadrant 2 and then increases quickly into quadrant 1.

The domain of an exponential function, such as \( y = b^x \) where \( b > 1 \), is all real numbers, which is represented as \( (-\infty, +\infty) \). This means that you can plug in any real number into the function for \( x \) and you will get a result.

The range of an exponential function is \( (0, +\infty) \). This tells us that the output of the function (the \( y \) values) will always be greater than 0, no matter what \( x \) value you choose. The function will never touch or cross the x-axis because the y-value can never be zero or negative.

When you graph \( y = 2^x \) on the coordinate plane, the curve will approach the x-axis as x goes to negative infinity, but will never actually touch it. This behavior is known as an asymptote, particularly a horizontal asymptote at \( y = 0 \). As x increases, the value of \( y \) will grow exponentially becoming very large as you move further to the right. In other words, the graph will increase rapidly in quadrant 1, which includes the positive x and y values.

Extra: The exponential function \( y = b^x \) has some important characteristics:

1. Exponential Growth: If \( b > 1 \), the function represents exponential growth, that is, as \( x \) increases, \( y \) increases rapidly. The larger the base \( b \), the steeper the growth.

2. Horizontal Asymptote: Exponential functions have a horizontal asymptote, usually represented by the x-axis (\( y = 0 \)). This means that the function never touches or crosses the x-axis; it just gets infinitely close to it.

3. Continuity: Exponential functions are continuous for all real numbers. There are no breaks, holes, or jumps in the graph of an exponential function.

4. End behavior: The end behavior of an exponential function tells us what the graph does as \( x \) goes to negative or positive infinity. In the case of \( y = 2^x \), as \( x \) approaches negative infinity, \( y \) approaches 0. As \( x \) approaches positive infinity, \( y \) becomes increasingly large.

5. Exponential functions also have a unique point known as the y-intercept, which is the point at which the curve crosses the y-axis. Since any number raised to the power of 0 is 1, for all exponential functions of the type \( y = b^x \), the y-intercept is at (0, 1).

Understanding these characteristics can help students graph exponential functions by hand and also better understand their properties when studying growth and decay processes in fields such as biology, economics, and physics.