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2-1 Power and Radical Functions Part 1 Chapter 2 Power, Polynomial, and Rational Functions Objectives You have already analyzed parent functions and their families of graphs. Now we will be Graphing and analyzing power functions, Graphing and analyzing radical functions, and (Solving radical equations next class) 1. Power functions Power functions previously studied are f ( x) x A power function is any function of the form 2 f ( x ) ax n where a and n are nonzero constant real numbers. and f ( x) x 3 End Behavior: using f(x)= axn When a > 0 When n is even When n is odd When a < 0 As x –∞, As x –∞, As x ∞, As x ∞, As x –∞, As x –∞, As x ∞, As x ∞, We should be able to also identify domain, range, intercepts, continuity, and where the function is increasing and decreasing. Example 1a: f ( x) 1 x 6 2 Graph and analyze the function. Describe the domain, range, intercepts, end behavior, continuity, and where the function is increasing or decreasing. Example 1b: f ( x) x 5 Graph and analyze each function. Describe the domain, range, intercepts, end behavior, continuity, and where the function is increasing or decreasing. 2. Functions with Negative Exponents Recall that f ( x ) 1 or x 1 is undefined at x = 0. x 2 Similarly, f ( x ) x and f ( x ) x 3 are undefined at x = 0. Because power functions can be undefined when n < 0 (negative exponents), these functions will have discontinuities. Example 2a: f ( x) 2 x 4 Graph and analyze the function. Describe the domain, range, intercepts, end behavior, continuity, and where the function is increasing or decreasing. Example 2b: f ( x ) 2 x 3 Graph and analyze the function. Describe the domain, range, intercepts, end behavior, continuity, and where the function is increasing or decreasing. 3. Functions with Rational Exponents 1 Recall that f ( x ) x n is a root function. p f ( x ) x n indicates the nth root of xp p If n is an even integer, then the domain must be restricted to f (x) x n nonnegative values. 3 Example 3a: f (x) 2 x 4 Graph and analyze the function. Describe the domain, range, intercepts, end behavior, continuity, and where the function is increasing or decreasing. 5 Example 3b: f ( x ) 10 x 3 Graph and analyze the function. Describe the domain, range, intercepts, end behavior, continuity, and where the function is increasing or decreasing. 4. Graphing Radical Functions An expression with rational exponents can be written in radical form. Exponential form vs. Radical form p f (x) x n f ( x) n x p Example 4a: f ( x) 5 2 x 3 Graph and analyze the function. Describe the domain, range, intercepts, end behavior, continuity, and where the function is increasing or decreasing. Example 4b: f ( x) 1 5 3x 4 2 Graph and analyze the function. Describe the domain, range, intercepts, end behavior, continuity, and where the function is increasing or decreasing. Assignment: p. 92 1, 5, 9, 13,19, 23, 27, 35, 37 2-1 Power and Radical Functions Part 2 Chapter 2 Power, Polynomial, and Rational Functions Warm up Graph and analyze the function. Describe the domain, range, intercepts, end behavior, continuity, and where the function is increasing or decreasing. 5 1. g ( x) 2. h( x) 3 x x 8 8 7 3. f ( x ) 3 6 3 x Answer: 1. g ( x) 5 8 x 8 Answer: 2. h( x) 3 x 7 Answer: 3. f ( x) 3 6 3x Questions on the homework? Objectives: Learn how to use a graphing calculator to find a power function to model a set of data. 2. Solve radical equations. 1. 3. Homework quiz over last night’s homework. 4. Power Regression The table shows the braking distance in feet at several speeds in miles per hour for a specific car on a dry, well-paved roadway. Speed 10 20 30 40 50 60 70 Distance 4.2 16.7 37.6 66.9 104.5 150.5 204.9 If we wanted to find a model to represent this relationship, a graphing calculator should make this an easy task. Write these steps down in your notes to refer to later, when you need to do another model. Speed 10 20 30 40 50 60 70 Distance 4.2 16.7 37.6 66.9 104.5 150.5 204.9 1. Enter the data in L1 (speed) and L2 (distance) (To access, press STAT, then EDIT. Clear the old lists but DO NOT DEL(ete). 2. Create a scatter plot of the data (2nd STATPLOT, Select Plot 1, Turn it on [if it is off]. Select a scatter plot, and the appropriate lists. Press ZOOM 9) 3. If the plot appears to be a power function, use the use the STAT, CALC, PwrReg function. (To place this equation in the Y= menu, go to Y1=, then VARS, Statistics, EQ, RegEQ) 4. Check your scatter plot again. The regression line should appear on the screen with the data points. 5. Use the equation to make a prediction of the braking distance of a car going 80 miles per hour. A table of values starting at 79 would work just fine. Solving Radical Equations First, isolate the radical expression. Then, raise each side of the equation to a power equal to the index of the radical to eliminate the radical. Note: This process sometimes produces extraneous solutions, which are solutions which do not satisfy the original equation. Always check your solutions in the original equation to determine if extraneous solutions exist. Example 1. 2x 28 x 29 3 Example 2. 12 3 ( x 2) 8 2 Example 3. x 1 1 2 x 12 Assignment: p. 92-93, 33, 45 - 55 odds, 100 - 103.