1. 1.5 Graphical Transformations
Represent translations algebraically and graphically

2. Consider this…
How is the graph (x – 2)2 + (y+1)2 = 16 related to the graph of x2 + y2 = 16?

3. Some change is good!! 
Transformations - functions that map real numbers to real numbers
Rigid transformations – leave the size and shape of a graph unchanged, include horizontal translations, vertical translations, reflections or any combination of these.
Non-rigid transformations – generally distort the shape of a graph, include horizontal or vertical stretches and shrinks.

4. Vertical and Horizontal Translations
1.5 Graphical Transformations
Vertical and Horizontal Translations
Vertical translation – shift of the graph up or down in the coordinate plane
Horizontal translation – shift of the graph left or right in the coordinate plane

5. Exploration #1 Complete the activity on p. 132
No talking – first 4 min.
You will be able to discuss with classmates the last 2 min.

6. Translations
Let c be a positive real number. Then the following transformations result in translations of the graph of y = f(x)
Horizontal translations
y = f(x – c) a translation to the right by c units
y = f(x + c) a translation to the left by c units
Vertical translations
y = f(x) + c a translation up by c units
y = f(x) – c a translation down by c units

7. Ex 1 Describe the graph of y = |x| can be transformed to the graph of the given function:
a) y = |x – 4| b) y = |x| + 2

8. Reflections, Stretches, and Shrinks
Represent reflections, stretches, and shrinks of functions algebraically and graphically

9. Graph in the Mirror!! 
Reflections – the graphs of two functions are symmetric with respect to some line
Complete Exploration #2 on p. 134
First 6 min (No Talking)
Last 2 min (Discuss with a neighbor)

10. Reflections
Over the x-axis – flips the graph of a function over the x-axis
Symbolically (x,y)  (x,-y)
Over the y-axis – flips the graph of a function over the y-axis
Symbolically (x,y)  (-x,y)
Over the line y = x – flips the graph of a function over the line y = x
Symbolically (x,y) (y,x)

11. Ex 1 Find an equation for the reflection of across each axis

12. Tonight’s Assignment
P. 139 – 140 Ex # 3-24 m. of 3

13. Ex: Express h(x) so that it represents the graph of f(x) = x2 – 3 reflected over the x-axis? y-axis?

14. Stretching & Shrinking
Complete the exploration on p. 136
First 8 min. No talking
Last 8 min. you can discuss with a neighbor

15. Stretches and Shrinks
Let c be a positive real number. The following transformations result in stretches or shrinks of the graph of y = f(x).
Horizontal stretches or shrinks
y = f(x/c) a stretch by a factor of c if c > 1
a shrink by a factor of c if c < 1
Vertical stretches or shrinks
y = cf(x) a stretch by a factor of c if c > 1

16. a) f(x) = |x + 2| b) f(x) = x2 + x - 2
Ex 3 Transform the given function by (a) a vertical stretch by a factor of 2 and (b) horizontal shrink by a factor of 1/3.
a) f(x) = |x + 2| b) f(x) = x2 + x - 2

17. Combining Transformations
The order in which transformations are performed often affect the graph that results
Ex 4 Use f(x) = x2 to perform each transformation. Write the formula for the resulting function.
A horizontal shift 2 units to the right, a vertical stretch by a factor of 3, and vertical translation 5 units up
Apply the transformations in the reverse order
Are the graphs the same? Are the formulas the same?