1. Evaluating Functions

2. Example Remember this example… If g(x) = x2 + 2x, evaluate g(x – 3)
g(x-3) = (x2 – 6x + 9) + 2x - 6
g(x-3) = x2 – 6x x - 6
g(x-3) = x2 – 4x + 3
What does this mean?

3.   g(x) = x2 + 2x g(x-3) = x2 – 4x + 3 x = -b = 4 x = -b = -2 = 22a
x = -b 2a
= 4
2(1)
= -2
2(1)
= 2
= -1
y = (2)2 – 4(2) + 3
= -1
y = (-1)2 + 2(-1)
= -1  
Vertex =
(-1,-1)
Vertex =
(2,-1)
Pattern =
1,3,5
Pattern =
1,3,5

4. This leads us into transformations…
Once you know f(x), then
f(x) + c
f(x) – c
f(x + c)
f(x - c)
all indicate a transformation.

5. There are two kinds of transformations:
Rigid
Non-rigid

6. Vocabulary:
Rigid Transformation – a shift, slide or reflection of a graph.
Non-rigid Transformation – a distortion of a graph by vertical or horizontal stretching.

7. Rigid Transformations
f(x) + c
f(x) – c
f(x + c)
f(x - c)
-f(x)
f(-x)
All maintain the exact same shape of the graph. The graph is just
repositioned.

8. f(x) + c
Moves up c squares. (Adding c to all y’s)
f(x) - c
Moves down c squares. (Subtracting c from all y’s)
f(x + c)
Moves left c squares. (Subtracting c from all x’s)
f(x - c)
Moves right c squares. (Adding c to
all x’s)

9. -f(x)
Reflects over the x-axis .
f(-x)
Reflects over the y-axis.

10. Graph f(x-2) in a different color, but on the same grid.
Let’s try:
f(x) = x2
Graph it in pencil.
Graph f(x-2) in a different color, but on the same grid.
Graph f(x) – 4 in a different color.
Graph -f(x) in a different color.
Graph f(x + 3) + 1 in a different color.

11. Graph f(x-1) in a different color, but on the same grid.
Let’s try:
f(x) = |x|
Graph it in pencil.
Graph f(x-1) in a different color, but on the same grid.
Graph f(x) +2 in a different color.
Graph -f(-x) + 1 in a different color.
Graph -f(x - 1) in a different color.

12. Graph -f(x) in a different color, but on the same grid.
Let’s try:
f(x) = x
Graph it in pencil.
Graph -f(x) in a different color, but on the same grid.
Graph f(-x) -1 in a different color.
Graph f(x + 3) + 1 in a different color.
Graph -f(x) - 1 in a different color.

13. Graph f(x - 1) + 3 in a different color, but on the same grid.
Let’s try:
f(x) =
3x – 1, x > 0
x + 1, x < 0
Graph it in pencil.
Graph f(x - 1) + 3 in a different color, but on the same grid.
Graph -f(x + 2) in a different color.
Graph f(x) -3 in a different color.
Graph f(-x) in a different color.

14. Non - Rigid Transformations
f(nx)
nf(x)
These distort the shape of the graph.

15. non-rigid transformations work.
Let’s see how
non-rigid transformations work.
f(x) = |x|
2f(x)
f(2x) x y x y x y
Notice what happened to the y-value…

16. nf(x)
All the y’s of f(x) get multiplied by n.
f(nx)
All the x’s of f(x) get divided by n.

17. Graph f(2x) in a different color, but on the same grid.
Let’s try:
f(x) = x2
Graph it in pencil.
Graph f(2x) in a different color, but on the same grid.
Graph 2f(x) in a different color.
Graph 1/2f(x) in a different color.
Graph f(1/2x) in a different color.

18. Let’s create a unique shape and try one more time.