Exponential Functions and Their Graphs MATH Precalculus S. Rook

Overview Section 3.1 in the textbook: – Exponential functions – Graphing exponential functions – e x 2

Exponential Functions

Thus far we have discussed linear and polynomial functions There exist applications which cannot be modeled by linear or polynomial functions: – e.g. bacteria reproduction, amount of a radioactive substance, continuous compounding of a bank account In all of these cases, a value in the model changes by a multiple of the previous value – e.g. A population that starts with 2 members and doubles every hour 4

5 Exponential Functions (Continued) Exponential function: f(x) = a x where the base a > 0, a ≠ 1 and x is a real number – If the base were negative, some values of x would result in complex values To evaluate an exponential function – Substitute the value for x and evaluate the expression

Evaluating an Exponential Function (Example) Ex 1: Use a calculator to estimate: a)f(x) = 3.4x when x = 5.6 – round to three decimal places b)g(x) = 5 x when x = 2 ⁄ 3 – round to three decimal places 6

Graphing Exponential Functions

8 To graph an exponential function f(x) = a x, make a table of values: – If a > 1 and x > 0, we will get a curve something like that on the right – If 0 < a < 1 OR x < 0, we will get a curve something like that on the right

9 Properties of Exponential Functions, a > 0 and x > 0 Does f(x) = a x have an inverse? – Yes, any horizontal line will cross f(x) only once What happens when x = 0? a 0 = 1 → y-int: (0, 1) What is the domain and range of f(x)? Domain: (-oo, +oo) Range: (0, +oo) What can be noticed about the end behavior of f(x)? – As x  -oo, f(x)  0 & as x  +oo, f(x)  +oo

10 Properties of Exponential Functions, 0 < a < 1 or x < 0 Does f(x) = a -x or f(x) = a x (0 < a < 1) have an inverse? – Yes, any horizontal line will cross f(x) only once What happens when x = 0? a 0 = 1 → y-int: (0, 1) What is the domain and range of f(x)? Domain: (-oo, +oo) Range: (0, +oo) What can be noticed about the end behavior of f(x)? – As x  -oo, f(x)  +oo & as x  +oo, f(x)  0

Graphing Exponential Functions (Example) Ex 2: Use a calculator to obtain a table of values for the function and then sketch its graph: a)f(x) = 3 x b)g(x) = 6 -x 11

12 Exponential Functions & Transformations We can also apply transformations to graph exponential functions Recall the following types of transformations: – Horizontal and vertical shifts – Horizontal and vertical stretches & compressions – Reflections over the x and y axis

Exponential Functions & Transformations (Example) Ex 3: Use the graph of f to describe the transformation(s) that yield the graph of g a)f(x) = 3 x g(x) = 3 x – 4 – 2 b) 13

exex

15 exex e is a mathematical constant discovered by Leonhard Euler – Used in many different applications – Deriving the value of e is somewhat difficult and you will learn how to do so when you take Calculus Natural exponential function: f(x) = e x where e ≈ (a constant) We can graph e x by creating a table of values and we can also apply translations f(x) = e x

e x ( Example) Ex 4: Use a calculator to estimate f(x) = e x when x = 10 and when x = 7 ⁄ 4 – round to three decimal places 16

One-to-One Property

As previously discussed, exponential functions are one-to-one functions – One value of y for every x and vice versa One-to-one Property: If a > 0, a ≠ 1, a x = a y → x = y – i.e. Obtain the same base and equate the exponents 18

One-to-One Property (Example) Ex 5: Use the One-to-One Property to solve the equation for x: a)b) c) 19

Summary After studying these slides, you should be able to: – Understand the concept of an exponential function and the limitations on the base – Describe the graph of an exponential function by looking at the base – State the domain and range of an exponential function – Graph an exponential function using a table of values and translations – Understand the constant e and be able to graph the natural exponential function using a t-chart and translations Additional Practice – See the list of suggested problems for 3.1 Next lesson – Logarithmic Functions and Their Graphs (Section 3.2) 20