Essential Questions 1)What is the difference between an odd and even function? 2)How do you perform transformations on polynomial functions? Even and Odd Functions (graphically) If the graph of a function is symmetric with respect to the y-axis, then it’s even. If the graph of a function is symmetric with respect to the origin, then it’s odd. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 2 Even and Odd Functions (algebraically) A function is even if f(-x) = f(x) If you plug in -x and get the original function, then it’s even. A function is odd if f(-x) = -f(x) If you plug in -x and get the opposite function, then it’s odd. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 3 Let’s simplify it a little… We are going to plug in a number to simplify things. We will usually use 1 and -1 to compare, but there is an exception to the rule….we will see soon! Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 4 Ex. 1 Even, Odd or Neither? Graphically f ( x) x Algebraically f ( x) x f (1) 1 1 f (1) 1 1 They are the same, so it is..... Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 5 Ex. 2 Even, Odd or Neither? f ( x) x x Graphically 3 What happens if we plug in 1? Algebraically 3 f ( x) x x f (2) (2) (2) 6 3 f (2) (2) (2) 3 6 They are opposite, so… Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 6 Ex. 3 Even, Odd or Neither? f ( x) x 1 Graphically 2 Algebraically f ( x) x 1 2 f (1) (1) 1 2 2 2 f (1) (1) 1 2 They are the same, so..... Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 7 Ex. 4 Even, Odd or Neither? f ( x) x 1 Graphically 3 Algebraically f ( x) x 1 3 f (1) (1) 1 3 0 3 f (1) (1) 1 2 They are not = or opposite, so... Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 8 Let’s go to the Task…. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 9 What happens when we change the equations of these parent functions? Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 10 f ( x) ( x 9) 14 Left 9 , Down 14 f ( x) ( x 2) 3 Left 2 , Down 3 2 Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 11 What did the negative on the outside do? -f(x) Reflection in the x-axis Study tip: If the sign is on the outside it has “x”-scaped What do you think the negative on the inside will do? f(-x) Reflection in the y-axis Study tip: If the sign is on the inside, say “y” am I in here? Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 12 Write the Equation to this Graph y ( x 3) 2 2 Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 13 Write the Equation to this Graph y ( x 2) 1 3 Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 14 Write the Equation to this Graph y x 1 1 Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 15 Write the Equation to this Graph f ( x) ( x) 2 or f ( x) ( x) 2 3 Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 3 16 Example: Sketch the graph of f (x) = – (x + 2)4 . This is a shift of the graph of y = – x 4 two units to the left. This, in turn, is the reflection of the graph of y = x 4 in the x-axis. y y = x4 x f (x) = – (x + 2)4 Copyright © by Houghton Mifflin Company, Inc. All rights reserved. y = – x4 17 Compare: 1 3 f ( x) x and g ( x) 4 x and h( x) x 4 3 Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 3 18 Compare… f ( x) x to f ( x) 3x What does the “a” do? Compare… • What does the “a” do? Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 2 2 Vertical stretch 1 2 f ( x) x to f ( x) x 2 2 Vertical shrink 19 Nonrigid Transformations h(x) = c f(x) c >1 Vertical stretch Closer to y-axis 0<c<1 Vertical shrink Closer to x-axis Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 20 Polynomial functions of the form f (x) = x n, n 1 are called power functions. 5 f (x) = x 4 y f (x) = x y f (x) = x2 f (x) = x3 x If n is even, their graphs resemble the graph of f (x) = x2. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. x If n is odd, their graphs resemble the graph of f (x) = x3. 21