- 122 Views
- Uploaded on

Download Presentation
## PowerPoint Slideshow about ' Lesson 1-6, page 194' - decker

**An Image/Link below is provided (as is) to download presentation**

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.

- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -

Presentation Transcript

### Lesson 1-6, page 194

Symmetry & Transformations

1.6 Transformation of Functions

- Recognize graphs of common functions
- Use vertical shifts to graph functions
- Use horizontal shifts to graph functions
- Use reflections to graph functions
- Use vertical stretching & shrinking to graph functions
- Use horizontal stretching & shrinking to graph functions
- Graph functions w/ sequence of transformations

Basic Functions

- You should memorize the following basic functions. They are vital to understanding upper level math courses.
- See the back section of your book,
Algebra’s Common Functions, for all of the basic functions you should memorize. (See inside back cover and turn back 1 or 2 pages.)

Types of Transformations

- Vertical translation: shifts basic graph up or down; up if y = f(x) + b ordown if y = f(x) – b
- Horizontal translation: shifts basic graph left or right; right if y = f(x - d) orleft if y = f(x + d)
- Reflection over x-axis:
across x axis if y = -[f(x)]

- Vertical stretching and shrinking:y = af(x)
stretch if |a| > 1; shrink if 0<|a|<1

- Vertical shifts
- Moves the graph up or down
- Impacts only the “y” values of the function
- No changes are made to the “x” values

- Horizontal shifts
- Moves the graph left or right
- Impacts only the “x” values of the function
- No changes are made to the “y” values

Recognizing the shift from the equation, look at examples of shifting the function f(x)=x2.

- Vertical shift of 3 units up
- Horizontal shift of 3 units left (HINT: x’s go the opposite direction of what you might think.)

Combining a vertical & horizontal shift shifting the function f(x)=x

- Example of function that is shifted down 4 units and right 6 units from the original function.

Types of Symmetry shifting the function f(x)=x

Symmetry with respect to the

- y-axis (x, y) & (-x, y) are reflections across the y-axis
- Origin (x, y) & (-x, -y) are reflections across the origin
- x-axis(x, y) & (x, -y) are reflections across the x-axis

Tests of Symmetry shifting the function f(x)=x

- f(x) = f(-x) symmetric to y-axis
even function

(replace x with –x)

- - f(x) = f(-x) symmetric to origin
odd function

(replace x with –x & y with –y)

- f(x) = - f(x) symmetric to x-axis
neither even nor odd function

(replace y with –y)

**Note** Functions don’t have to be symmetric to the x axis to be

neither even nor odd. (ex. y=x^2+3x) not even or odd function

and also not symmetric to x axis.

Example – Determine any/all symmetry of shifting the function f(x)=x6x + 7y = 0.

a) y-axis (replace x w/ -x)

b) origin (replace both)

c) x-axis (replace y w/ -y)

- Is the function even, odd, or neither?

Example – Determine any/all symmetry of shifting the function f(x)=x

a) y-axis (replace x w/ -x)

b) origin (replace both)

c) x-axis (replace y w/ -y)

- Is the function even, odd, or neither?

***SKIP*** shifting the function f(x)=xHorizontal stretch & shrink

f(x) = |x2 – 4|

- We’re MULTIPLYING by an integer (not 1 or 0).
- x’s do the opposite of what we think they should. (If you see 3x in the equation where it used to be an x, you DIVIDE all x’s by 3, thus it’s compressed or shrunk horizontally.)

VERTICAL STRETCH (SHRINK) shifting the function f(x)=x

f(x) = |x2 – 4|

- y’s do what we think they should: If you see 3(f(x)), all y’s are MULTIPLIED by 3 (it’s now 3 times as high or low!)

Sequence of transformations shifting the function f(x)=x

- Follow order of operations.
- Select two points (or more) from the original function and move that point one step at a time.

Graph of Example shifting the function f(x)=x

Transformations with shifting the function f(x)=xthe Squaring Function

FunctionTransformation

- g(x) = x2 + 4 ________________________
- h(x) = x2 – 5 _______________________
- j(x) = (x – 3)2 ________________________
- k(x) = (x + 1)2 ________________________
- q(x) = -x2 ________________________
- r(x) = 2x2 ________________________
- s(x) = ¼x2________________________

Your turn. Describe these transformations with the Absolute Value Function.

FunctionTransformation

- g(x) = -|x| ________________________
- j(x) = 3|x|________________________
- k(x) = |x + 4|________________________
- r(x) = |x| - 1________________________
- s(x) = -½|x|________________________
- t(x) = |x| + 2 ________________________

Summary of Transformations Value Function.

- See table 1.5 on page 203.
- This is an excellent summary of all the transformations studied.

Download Presentation

Connecting to Server..