Essential Questions 1)What is the difference between an odd and even function? 2)How do you perform transformations on polynomial functions?

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Presentation transcript:

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. The easiest thing to do is to plug in 1 and -1 (or 2 and -2) if you get the same y, then it’s Even. If you get the opposite y, then it’s Odd. If you get different y’s, then it’s Neither. Copyright © by Houghton Mifflin Company, Inc. All rights reserved.

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) The easiest thing to do is to plug in 1 and -1 (or 2 and -2) if you get the same y, then it’s Even. If you get the opposite y, then it’s Odd. If you get different y’s, then it’s Neither. If you plug in -x and get the opposite function, then it’s odd. Copyright © by Houghton Mifflin Company, Inc. All rights reserved.

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.

EVEN Ex. 1 Even, Odd or Neither? Graphically Algebraically Copyright © by Houghton Mifflin Company, Inc. All rights reserved.

ODD Ex. 2 They are opposite, so… Even, Odd or Neither? Graphically What happens if we plug in 1? Graphically Algebraically ODD They are opposite, so… Copyright © by Houghton Mifflin Company, Inc. All rights reserved.

EVEN Ex. 3 Even, Odd or Neither? Graphically Algebraically Copyright © by Houghton Mifflin Company, Inc. All rights reserved.

Neither Ex. 4 Even, Odd or Neither? Graphically Algebraically Copyright © by Houghton Mifflin Company, Inc. All rights reserved.

Let’s go to the Task…. Copyright © by Houghton Mifflin Company, Inc. All rights reserved.

What happens when we change the equations of these parent functions? Copyright © by Houghton Mifflin Company, Inc. All rights reserved.

Describe the Shift Left 9 , Down 14 Left 2 , Down 3 Copyright © by Houghton Mifflin Company, Inc. All rights reserved.

-f(x) f(-x) Reflection in the x-axis Reflection in the y-axis What did the negative on the outside do? Reflection in the x-axis -f(x) 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.

Write the Equation to this Graph Copyright © by Houghton Mifflin Company, Inc. All rights reserved.

Write the Equation to this Graph Copyright © by Houghton Mifflin Company, Inc. All rights reserved.

Write the Equation to this Graph Copyright © by Houghton Mifflin Company, Inc. All rights reserved.

Write the Equation to this Graph Copyright © by Houghton Mifflin Company, Inc. All rights reserved.

Example: Graph of f(x) = – (x + 2)4 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. x y y = x4 f (x) = – (x + 2)4 y = – x4 Copyright © by Houghton Mifflin Company, Inc. All rights reserved. Example: Graph of f(x) = – (x + 2)4

Compare: Copyright © by Houghton Mifflin Company, Inc. All rights reserved.

Compare… Compare… What does the “a” do? What does the “a” do? Vertical stretch Compare… What does the “a” do? Vertical shrink Copyright © by Houghton Mifflin Company, Inc. All rights reserved.

Nonrigid Transformations Vertical stretch c >1 h(x) = c f(x) Closer to y-axis 0 < c < 1 Vertical shrink Closer to x-axis Copyright © by Houghton Mifflin Company, Inc. All rights reserved.

If n is even, their graphs resemble the graph of f (x) = x2. Polynomial functions of the form f (x) = x n, n  1 are called power functions. f (x) = x5 x y f (x) = x4 x y f (x) = x2 f (x) = x3 If n is even, their graphs resemble the graph of f (x) = x2. If n is odd, their graphs resemble the graph of f (x) = x3. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. Power Functions