Mrs. McConaughy rex Honors Algebra 2 1

The number of o-rings damaged increases dramatically as Fahrenheit temperature falls. The function f(x) = 13.49 (0.967)x -1 models the number of O-rings expected to fail when the temperature is x◦ F.

The space shuttle Challenger exploded 73 seconds into flight on January 28, 1986. The tragedy involved damage to o-rings , which were used to seal the connections between different sections of the shuttle engines.

msnbc.com

Mrs. McConaughy rex Honors Algebra 2 2

8.1 Exploring Exponential Models

Objectives:

To graph exponential functions To evaluate exponential expressions To use exponential functions as models for real-life problems

Mrs. McConaughy rex Honors Algebra 2 3

Exponential Growth and Decay Models

Earlier, we studied graphs of basic exponential functions of the form f(x) = C(a)x,

where 0<a<1 represents ___________ and

a>1 represents _______________.

Exponential Decay: f(x) = (1/2)x

Exponential Growth: f(x) = (3)x

(0,1)

(0,1)

decay

growth

Mrs. McConaughy rex Honors Algebra 2 4

In this lesson, we will study vertical shifts, horizontal shifts and reflections of these two basic types of graphs.

EXAMPLE 1 Using Vertical Shift to Sketch Graphs

(0,1) (0,1)

To sketch the graph of g, shift the graph f(x) = (1/2)x up one unit.

To sketch the graph of h, shift the graph f(x)= 3x down two units.

(0,2)

(0,-1)

Sketch g(x) = (1/2)x + 1Sketch h(x) = 3x - 2

Mrs. McConaughy rex Honors Algebra 2 5

Horizontal ShiftsEXAMPLE 2 Using Horizontal Shifts

y = 2x-2

y = 2x+3 y = 2x

To sketch the graph of g, shift f(x) = 2x to the left 3 units.To sketch the graph of h, shift f(x) = 2x to the right 2 units.

Graph f(x) = 2xSketch g(x) = 2x + 3Graph h(x) = 2x -2

Mrs. McConaughy rex Honors Algebra 2 6

g(x) = -(1/2)x + 4

Reflections and ShiftsEXAMPLE 3

a.Using Reflections and

Shifts

First, reflect the graph of f(x) = (1/2)x in thex axis.

Sketch the graph of g(x) = -(1/2)x + 4 :

f(x) = (1/2)x h(x) = - (1/2)x

Then shift the Graph upward 4 units.

(0,1) (0,3)

(0,-1)

Mrs. McConaughy rex Honors Algebra 2 7

Reflections and ShiftsEXAMPLE 3b.

Using Reflections and Shifts

Sketch the graph of h(x) = - 2x-2 + 2 :

f(x) = 2x h(x) = -2x-2 + 2

(0,1)

To sketch the graph of h, shift the graph of f(x) = 2x

2 units to the right. Then reflect the graph in the x-axis and shift 2 units upward.

Mrs. McConaughy rex Honors Algebra 2 8

Use a calculator to evaluate the power.Round the result to 3 decimal places.

a. 3√2

POWERS THAT HAVE IRRATIONAL EXPONENTS

EXAMPLE 4 Evaluating Powers that have Irrational

Exponents

b. 3 - Π

Mrs. McConaughy rex Honors Algebra 2 9

EXAMPLE 5

EXPONENTIAL DECAY MODEL

Suppose your parents purchased a new car for $20,000.00 in 2000. If the value of the car depreciated by 16% each year to 84% of its previous value, what was the car worth in 2006? What will it be worth in 2008?

Hint: Exponential Decay ModelV = Cat

= Initial Value (0.84)t

Mrs. McConaughy rex Honors Algebra 2 10

Final Checks for Understanding

1. Compare the graphs of f(x) = 5x and g(x) = 5x - 3. 1. Compare the graphs of f(x) = 3x and g(x) = -3 x +2. 1. Use a calculator to evaluate 5 √3. Round

the result to three decimal places.2. Which function represents the value after

x years of a delivery van that was purchased new for $17,500 and depreciates 11% every year?

A. y = -11 (17,500)x

B. y = 17,500 (0.11)x

C. y = 17,500 (.89)x

D. y = 17,500 (1.11)x

Mrs. McConaughy rex Honors Algebra 2 11

Homework Assignment:

Exponential Functions WS

Day 1: 5-11 odd (use complete sentences); 13-20 all (matching); 21, 25, 29, 33 (graphing); 40, 44 (calculator)