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Lesson 1-6, page 194

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Lesson 1-6, page 194

Symmetry & Transformations

- 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

- 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.)

- 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

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

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

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

- 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.

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?

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?

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.)

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!)

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

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________________________

FunctionTransformation

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

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