Graphing Exponential and Logarithmic Functions: Tutorial

From the graph of f(x) = 2^x, we can find this information:

- Intercepts—Notice the point where the graph intersects the y-axis at A(0, 1). This is the y-intercept. The graph does not intersect the x-axis at any point, so there is no x-intercept. - Asymptote—As x decreases, the value of f(x) decreases. For lower values of x, the graph descends and comes closer to the x-axis but does not touch it. You can see that the horizontal asymptote of this function is the x-axis, or y = 0. - End behavior—As x increases, the value of f(x) decreases. For very low negative x-values, the graph moves closer to the asymptote (x-axis, or y = 0). As x approaches positive infinity, the graph increases positively toward infinity. Finding this information from f(x) = 2^x would normally involve manipulating the function in various ways. Because we graphed the function, we can determine this information by simply looking at the graph.

If you're given the algebraic form of an exponential function, you can find various features of the function. However, you can find this information more easily from the function's graph. Graphs reveal many behaviors of functions that are not immediately obvious when looking at the equations for these functions. You can identify these features of a function by examining its graph:

- the shape of the function - the end behavior of the function (its behavior at extreme values) - the domain and range of the function - the x-intercept (x, 0), where the graph intersects the x-axis - the y-intercept (0, y), where the graph intersects the y-axis - the asymptote (a line that the graph moves toward but does not intersect for a certain range of x-values)

Identify the end behavior of the given functions. The function f(x) = 2^x + 1 _____________ toward infinity as x increases. The function f(x) = 2^x-2 decreases toward ___________ as x decreases. The function f(x) = (1/3)^x-1 - 2 _______________ toward ____________ as x increases.

1. increases, infinity 2. decreases, the line y = 0 3. decreases, the line y = -2

Vertical and Horizontal Shifts

A horizontal or vertical shift in the graph of a logarithmic function does not change the shape of the graph, but it does result in a vertical or horizontal shift in the location of the graph. You've graphed functions of the form f(x) = log(x ± h) ± k with a horizontal shift by h units and a vertical shift by k units, where h and k are integers. Let's recap what you can infer from the graph of such a function. Consider the function f(x) = log(x + 4) + 2, which represents a horizontal shift by 4 units and a vertical shift by 2 units. The domain of this function is the set of real numbers starting at -4. Its range is the set of all real numbers. You can get this information from the graph: - Asymptote—As x decreases, f(x) decreases and comes closer to the vertical asymptote at x = -4. - Intercepts—The graph intersects the x-axis at -4, so the x-intercept is the point Q(-4, 0). The graph intersects the y-axis at 2.6020, so the y-intercept is the point P(0, 2.6020). - End behavior—As x increases, y approaches positive infinity. As x decreases, the graph moves toward the vertical asymptote.

An exponential function takes the form y = bx. What happens when b > 1? Let's see.

Consider the exponential function y = 2x. Here, b = 2 (b > 1). When x = 1, y = 2; when x = 2, y = 4; and when x = 3, y = 8. Every time x grows by 1, y doubles. A function where y grows with a growth in x is an exponential growth function. Population and compound interest, which both follow a pattern of growth over time, are examples of exponential growth functions. What if b < 1 (but b ≠ 0)? Consider the exponential Fiction y = (1/2)^x. Here, b= 1/2 (b < 1). When x = 1, y = (1/2)^1 = 1/2. When x = 2, y = (1/2)^2 = 1/4. When x = 3, y= (1/2)^3 = 1/8. Every time x grows by 1, y decreases to half its previous value. A function where y decreases with an increase in x is an exponential decay function. Examples of exponential decay functions include radioactive decay and decay of organic matter.

So far, we have discussed only positive exponential functions. However, exponential functions can also be negative. In negative exponential functions, the exponential value is multiplied by a negative number. Consider the function f(x) = -2(3)^x. The graph of this function (graph 1) is the graph of f(x) = 2(3)^x (graph 2) reflected about the x-axis. The information we can find from the graph of f(x) = -2(3)^x is listed in the table.

Feature 1. domain 2. range 3. x-intercept 4. y-inercept 5. horizontal asymptote Behavior 1. set of all real numbers 2. set of all negative real numbers 3. none 4. (0,2) 5. y = 0 End behavior - x approaching negative infinity - graph approaches horizontal asymptote - x approaching positive infinity - graph moves towards negative infinity

Graphing Logarithmic Functions

If you're given an exponential equation, b^x = y, and you know the values of b and y, you can find the value of x. You know that x is the exponent to which the base b is raised to get y. To find x, you can use logarithms, which is another way of expressing an exponent: b^x = y is the equivalent of logb y = x. For example, 25 = 32; so log2 32 = 5. Any positive number except 1 can be the base of a logarithm. The most commonly used bases are 10, also called the common base and written log(y), and base e, also called the natural base, which is equal to 2.718 . . . and written ln(y). You'll want to remember these points about logarithms: - Logarithmic values can be negative: 2^2 = 1/2, so log2 1/2 = -2. - Log 1 = 0 (because 10^0 = 1). - Logarithms of 0 and negative numbers are not defined. - You can find the base 10 logarithmic values of any positive number by passing the "log" button on a scientific calculator.

Vertical and Horizontal Shifts

It is possible to shift the graph of an exponential function horizontally or vertically by changing its equation. If an input value (the x) in a basic exponential function changes by h units (where h is an integer), the function shifts horizontally by h units. And the function changes form from y = b^x to y = b^x±h. If an integer k is added to or subtracted from a basic exponential function, the function shifts vertically by k units. And the function changes form from y = b^x to y = b^x ± k. Some functions can be shifted both horizontally and vertically. These functions take the form f(x) = b^x±h ± k. Horizontal and vertical shifts do not change the basic shape of an exponential function, but they do change the function's features.

Let's graph the exponential function f(x) = 2^x. Then we'll find its intercepts and asymptote and examine the function's end behavior. This is an exponential growth function because its base (b) is greater than 1.

To graph the function, we first create a table of different positive and negative x-values and the corresponding values of f(x). Then we plot the points and connect them to get the graph. Now we can study the graph to learn more about the function. For f(x) = 2x, x can take any real value, so the domain of this function is the set of all real numbers. For any real value of x, y is a positive value. That means the function's range is the set of all positive real numbers. We can confirm this fact from the graph. The graph lies only in the first and second quadrants, where y is positive for both positive and negative x-values.

You've graphed functions of the form f(x) = b^x±h ± k, which shift both horizontally and vertically. Let's summarize what happens to the graph of such a function. The exponential function f(x) = 3^x-1 + 1 has both horizontal and vertical shifts. The graph of this function (graph 2) is the graph of f(x) = 3^x (graph 1) shifted vertically by +1 unit and horizontally to the right by 1 unit.

We can find this information from the graph: - Asymptote—Because of the vertical shift, the horizontal asymptote moves up 1 unit to y = 1. - Intercepts—The y-intercept is (0, 1.33). The graph has no x-intercept. - End behavior—For very high x-values, the graph moves toward positive infinity. For very low negative x-values, the graph approaches the asymptote y = 1.

Graphing Exponential Functions

We use exponential functions for many purposes, such as calculating population growth and figuring out compound interest for mortgage payments. The basic form of an exponential function is y = b^x, where b is the base, b ≠ 0 and b ≠ 1, and x is the exponent. But how does the value of b affect how the exponential function changes with differing values of x? If b is large, does the function behave differently than it would if b were small? Let's look at some examples to better understand exponential functions with different base values.

Negative Logarithmic Functions

What happens when a log function is multiplied by a negative integer? How is the graph affected? Let's take the function f(x) = -log(x - 2) as an example. This function is the negative of the function f(x) = log(x - 2). The graph of the negative function f(x) = -log(x - 2) (graph 1) is the reflection of the graph of the positive function f(x) = log(x - 2) (graph 2) across the x-axis. The domain of the negative function f(x) = -log(x - 2) is the set of all positive real numbers greater than 2. Its range is the set of all real numbers. In addition, you can get this information from the graph: - Intercepts—The x-intercept is at (3, 0). There is no y-intercept. - Asymptote—As x decreases, f(x) decreases and moves closer to the vertical asymptote at x = 2. - End behavior—As the x-value increases beyond 3, the y-value decreases toward negative infinity. As the x-value decreases from 3 to 2, the graph moves toward the vertical asymptote.

You know that only positive values have logarithms. That means the domain of a basic logarithmic function is the set of all positive real numbers. Logarithmic values can be positive, negative, or zero. So, the range of a logarithmic function is the set of all real numbers. Notice that the graph of f(x) = log x is entirely in the first and fourth quadrants.

You can also determine these features from the graph: - Intercepts—The graph intersects the x-axis at (1, 0), which is the x-intercept, marked P in the graph. For smaller values of x, the graph moves closer to the y-axis but does not intersect it: there is no y-intercept. - Asymptote—As seen in this graph, logarithmic functions have vertical asymptotes. Here, for very small x-values, the graph comes close to but does not touch the y-axis: the y-axis (x = 0) is the asymptote. - End behavior—As x increases, y increases, and the graph moves toward positive infinity. For very low x-values, the graph approaches the vertical asymptote.