1. I can Shift, Reflect, and Stretch Graphs

2. Greatest Integer Function
||x|| is the greatest integer less than or equal to x So if f(x) = ||x|| f(3) = 3 f(4.3) = 4 f(5.9) = 5 f(-2.1) = -3 Now graph this!

3. Can you think of a real-life example?

4. Make sure that you know these basic functions and their graphs…
f(x) = c constant
f(x) = x identity
f(x) = x2 quadratic
f(x) = x3 cubic
f(x) = |x| absolute value
f(x) = √x square root
f(x) = 1/x reciprocal
f(x) = ||x|| greatest integer

5. Vertical and Horizontal Shifts
f(x) + c c units up (after )
f(x) – c c units down (after)
f(x + c) c units left (before)
f(x – c) c units right (before)
Ex: y = x2 + 2, y = x2 – 2, y = (x + 2)2, y = (x – 2)2

6. Reflections Reflection in the x-axis - f(x) (after)
Reflection in the y-axis
f(-x) (before)
Example 1: y = |x|, y = -|x|, y = |-x|
Example 2: y = x2 – 3, y = -(x2 – 3), y = (-x)2 – 3
Example 3: y = √x, y = -√x, y = √-x

7. Stretches
c •f(x) is a vertical expansion if c > 1 is a vertical compression if 0< c < 1 Example : y = 3x2, y = ½x2 f(c●x) is a horizontal compression if c> 1 is a horizontal expansion if 0< c < 1 Example: y = (4x)2, y = (.2x)2

8. Putting It All Together
It is very important to think about the order of the transformations…
Graph these functions
y = -(x + 3)2 + 1
y = 2|x - 4|
y = 4 - (x – 2)3
y = 1/2√x - 5
y = 1/(x+ 1) + 2

9. Piecewise Functions Graph f(x) = { x if -2 ≤ x≤ 1

10. Determine the transformation(s)
You will need to complete the square to put each quadratic in vertex form.
y= x2 – 4x + 7
right 2, up 3
y = -2x2 – 4x – 2
left one, stretched by a factor of 2, reflected over x-axis

11. Transformations with Absolute Value
|f(x)| reflects any part of f(x) below the x-axis over the x-axis f(|x|) replaces any part of the graph to the left of the y-axis with a reflection of the part of the graph to the right of the y-axis Think about y = x and y = |x|. Now do this for y = x – 3, y = |x - 3|, and y = |x|-3.

12. Graph y = |x2-2|. Graph y = x3 – 4x
Graph y = |x2-2|. Graph y = x3 – 4x. Now graph y = | x3 – 4x | and y = |x|3 - 4|x|.

13. THE END