3.1 EXPONENTIAL & LOG FUNCTIONS TRIG / PRE-CALC

Key Concept 2

Compare the graphs 2x, 3x , and 4x Characteristics about the Graph of an Exponential Function where a > 1 1. Domain is all real numbers 2. Range is positive real numbers 3. There are no x intercepts because there is no x value that you can put in the function to make it = 0 4. The y intercept is always (0,1) because a 0 = 1 5. The graph is always increasing 6. The x-axis (where y = 0) is a horizontal asymptote for x  - 

Key Concept 2

1. Domain is all real numbers   Characteristics about the Graph of an Exponential Function where a < 1 1. Domain is all real numbers 2. Range is positive real numbers 3. There are no x intercepts because there is no x value that you can put in the function to make it = 0 4. The y intercept is always (0,1) because a 0 = 1 5. The graph is always decreasing 6. The x-axis (where y = 0) is a horizontal asymptote for x  - 

Example 2 Graph Transformations of Exponential Functions A. Use the graph of to describe the transformation that results in . Then sketch the graph of the function. This function is of the form g (x) = f (x + 1). Therefore, the graph of g (x) is the graph of translated 1 unit to the left.

Example 2 Graph Transformations of Exponential Functions Answer: g (x) is the graph of f (x) translated 1 unit to the left.

Example 2 Graph Transformations of Exponential Functions B. Use the graph of to describe the transformation that results in . Then sketch the graph of the function. This function is of the form h (x) = f(–x). Therefore, the graph of h (x) is the graph of reflected in the y-axis.

Example 2 Answer: h (x) is the graph of f (x) reflected in the y-axis. Graph Transformations of Exponential Functions Answer: h (x) is the graph of f (x) reflected in the y-axis.

Example 2 Graph Transformations of Exponential Functions Answer: j (x) is the graph of f (x) reflected in the x-axis and expanded vertically by a factor of 2.

The Base “e” (also called the natural base) To model things in nature, we’ll need a base that turns out to be between 2 and 3. Your calculator knows this base. Ask your calculator to find e1. You do this by using the ex button (generally you’ll need to hit the 2nd or yellow button first to get it depending on the calculator). After hitting the ex, you then enter the exponent you want (in this case 1) and push = or enter. If you have a scientific calculator that doesn’t graph you may have to enter the 1 before hitting the ex. You should get 2.718281828

Graph of Natural Exponential Function f(x) = ex The graph of f(x) = ex y 6 x f(x) -2 0.14 -1 0.38 1 2.72 2 7.39 4 2 x –2 2 Graph of Natural Exponential Function f(x) = ex

Key Concept 3