Section 4 1 Exponential Functions Prepared by Mr

Section 4. 1 Exponential Functions Prepared by Mr. Awad

Objectives ■ Exponential Functions ■ Graphs of Exponential Functions

Definition of Exponential Function The exponential function is: with base where and is any real number, Domain is: “all real numbers” Note: In Math 001, x was rational exponent, here x is any real number. Prepared by Mr. Awad

The Base “a ” Base can’t be Negative, is an irrational number, e is an irrational number, Prepared by Mr. Awad

Example 1 Graph the exponential function: Prepared by Mr. Awad

Example 1: Solution Since this is a new function, generate some points: x y Prepared by Mr. Awad

Properties of Exponential Function *Domain: * Range: “ The set of all real numbers “ “without shifting” * Increasing “Base >1” * y-intercept: * it is one-to-one, so it has inverse (section 2. 8) • x-axis is horizontal asymptote ( Prepared by Mr. Awad )

Properties of Exponential Function *Domain: “ The set of all real numbers “ “without shifting” * Range: * Decreasing, Base: * y-intercept: * it is one-to-one, so it has inverse (section 2. 8) * x-axis is horizontal asymptote ( Prepared by Mr. Awad )

Example 2 Graph the following exponential functions on the same axes: Prepared by Mr. Awad

Example 2: Solution Notice that, as the base gets larger, the graph is closer to y-axis and x-axis: Prepared by Mr. Awad

Shifting Exponential Functions To graph exponential functions: (a) Graph first the basic exponential function without shifting. (b) Use reflection, (in the x-axis or in the y-axis). (c) Use shifting Vertical (Up or Down) and/or Horizontal (Right or Left). Prepared by Mr. Awad

Example 3 Graph the exponential functions: Find the Range of each. Prepared by Mr. Awad

Example 3 a): Solution Base is +3 > 1 “Increasing” Reflection in x-axis Prepared by Mr. Awad Shifting: 2 units left.

Example 3 b): Solution Basic is exponential base ½ < 1 “Decreasing” Prepared by Mr. Awad Shifting: 5 units up.

Example 4, “Very Important” “Exponential Functions with Absolute Value ” Graph the exponential functions: Find the Range of each. Prepared by Mr. Awad

Example 4 1): Solution Note: If absolute value is about y “whole function”, draw the function first then move all the negative part “below x-axis” TO “above x-axis” Prepared by Mr. Awad

Example 4 2): Solution Note: If absolute value is about x Prepared by Mr. Awad ”Take 2 cases”

Try at Home (1) Graph the functions, and find the Range: Prepared by Mr. Awad

Try at Home (1): Answers Part (a) Part (b) Prepared by Mr. Awad

Example 5 Given the exponential function: Find the x-intercept and y-intercept Prepared by Mr. Awad

Example 5: Solution x-intercept: y-intercept Prepared by Mr. Awad

Try at Home (2) If (a, 0) is the x-intercept, and (0, b) is the y-intercept of the graph: Find Prepared by Mr. Awad

Try at Home (2): Solution x-intercept: y-intercept Prepared by Mr. Awad

Example 6 If the function has the given graph, then: is equal to: Prepared by Mr. Awad

Example 6: Solution Given: Horizontal Asy. Point (0, 6) lies on the graph, the coordinates satisfy the equation of the function Same thing for the point (1, 0), Prepared by Mr. Awad

Try at Home (3) If is an exponential function of the form and then find Prepared by Mr. Awad

Try at Home (3): Solution Given: Since is a point on the graph. Now, replace x=-2, and y=1/3 to find a: But Base can’t be negative, Prepared by Mr. Awad

Practice # 1 Graph and find the Range of each exponential function: Prepared by Mr. Awad

Practice # 2 (Important) Graph the exponential function: then find: * * * Domain * Range * Image of 1 x-intercept * y-intercept Horizontal Asymptote. Where the graph is below x-axis. Where the graph is above x-axis. Prepared by Mr. Awad