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 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. Mrs. McConaughy rex Honors Algebra 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

Exponential Growth and Decay Models Exponential Decay: f(x) = (1/2)x 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 _______________. (0,1) Exponential Growth: f(x) = (3)x decay (0,1) growth Mrs. McConaughy rex Honors Algebra 2

Using Vertical Shift to Sketch Graphs 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 Sketch h(x) = 3x - 2 Sketch g(x) = (1/2)x + 1 (0,2) (0,1) (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. Mrs. McConaughy rex Honors Algebra 2

Using Horizontal Shifts EXAMPLE 2 Using Horizontal Shifts Sketch g(x) = 2x + 3 Graph h(x) = 2x -2 Graph f(x) = 2x y = 2x-2 y = 2x+3 y = 2x To sketch the graph of h, shift f(x) = 2x to the right 2 units. To sketch the graph of g, shift f(x) = 2x to the left 3 units. Mrs. McConaughy rex Honors Algebra 2

Reflections and Shifts EXAMPLE 3 a. Using Reflections and Shifts Sketch the graph of g(x) = -(1/2)x + 4 : g(x) = -(1/2)x + 4 f(x) = (1/2)x h(x) = - (1/2)x (0,1) (0,3) (0,-1) Then shift the Graph upward 4 units. First, reflect the graph of f(x) = (1/2)x in the x axis. Mrs. McConaughy rex Honors Algebra 2

Reflections and Shifts 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. EXAMPLE 3b. Using Reflections and Shifts Sketch the graph of h(x) = - 2x-2 + 2 : (0,1) h(x) = -2x-2 + 2 f(x) = 2x Mrs. McConaughy rex Honors Algebra 2

POWERS THAT HAVE IRRATIONAL EXPONENTS EXAMPLE 4 Evaluating Powers that have Irrational Exponents Use a calculator to evaluate the power. Round the result to 3 decimal places. a. 3√2 b. 3 - Π Mrs. McConaughy rex Honors Algebra 2

EXPONENTIAL DECAY MODEL 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 Model V = Cat = Initial Value (0.84)t Mrs. McConaughy rex Honors Algebra 2

Final Checks for Understanding Compare the graphs of f(x) = 5x and g(x) = 5x - 3. Compare the graphs of f(x) = 3x and g(x) = -3 x +2. Use a calculator to evaluate 5 √3. Round the result to three decimal places. 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

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) Mrs. McConaughy rex Honors Algebra 2