Essential Questions 1What is the difference between an
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… n 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 Copyright © by Houghton Mifflin Company, Inc. All rights reserved. Algebraically 5
Ex. 2 Even, Odd or Neither? Graphically Algebraically What happens if we plug in 1? They are opposite, so… Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 6
Ex. 3 Even, Odd or Neither? Graphically Copyright © by Houghton Mifflin Company, Inc. All rights reserved. Algebraically 7
Ex. 4 Even, Odd or Neither? Graphically Copyright © by Houghton Mifflin Company, Inc. All rights reserved. Algebraically 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
Left 9 , Down 14 Left 2 , Down 3 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 Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 13
Write the Equation to this Graph Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 14
Write the Equation to this Graph Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 15
Write the Equation to this Graph Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 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 = x 4 x f (x) = – (x + 2)4 Copyright © by Houghton Mifflin Company, Inc. All rights reserved. y = – x 4 17
Compare: Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 18
Compare… n What does the “a” do? Vertical stretch Compare… • What does the “a” do? Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 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 2 f (x) = x 3 x If n is even, their graphs resemble the graph of f (x) = x 2. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. x If n is odd, their graphs resemble the graph of f (x) = x 3. 21
- Slides: 21
