Remember this example…

ExampleIf g(x) = x2 + 2x, evaluate g(x – 3)

g( ) = 2 + 2x x(x -3) (x -3)g(x-3) = (x2 – 6x + 9) + 2x - 6

g(x-3) = x2 – 6x + 9 + 2x - 6g(x-3) = x2 – 4x + 3

What does this mean?

x -3x

g(x) = x2 + 2x g(x-3) = x2 – 4x + 3

                         

                         

                         

                         

                         

                         

                         

                         

                         

                         

                         

x = -b 2a

= -2 2(1)

= -1 x = -b 2a

= 4 2(1)

= 2

y = (-1)2 + 2(-1) = -1

Vertex = (-1,-1)

Pattern = 1,3,5

y = (2)2 – 4(2) + 3 = -1

Vertex = (2,-1)

Pattern = 1,3,5

This leads us into This leads us into transformations…transformations…

Once you know f(x), thenOnce you know f(x), thenf(x) + cf(x) + cf(x) – cf(x) – cf(x + c) f(x + c) f(x - c)f(x - c)

all indicate a all indicate a transformation.transformation.

There are two kinds of There are two kinds of transformations:transformations:

RigidRigid

Non-rigidNon-rigid

Vocabulary:Vocabulary:

Rigid Transformation Rigid Transformation – a – a shift, slide or reflection of a shift, slide or reflection of a graph.graph.

Non-rigid Transformation Non-rigid Transformation – – a distortion of a graph by a distortion of a graph by vertical or horizontal vertical or horizontal stretching.stretching.

Rigid TransformationsRigid Transformationsf(x) + cf(x) + cf(x) – cf(x) – cf(x + c) f(x + c) f(x - c)f(x - c)-f(x)-f(x)f(-x)f(-x)

All maintain the exact same All maintain the exact same shape of the graph. The shape of the graph. The graph is just graph is just repositioned.repositioned.

f(x) + cf(x) + cMoves up c squares. Moves up c squares. (Adding c to all y’s)(Adding c to all y’s)

f(x) - cf(x) - cMoves down c squares. Moves down c squares. (Subtracting c from all y’s)(Subtracting c from all y’s)

f(x + c)f(x + c)Moves left c squares. Moves left c squares. (Subtracting c from all x’s)(Subtracting c from all x’s)

f(x - c)f(x - c)Moves right c squares. Moves right c squares. (Adding c to(Adding c to all all x’s)x’s)

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

Let’s try:Let’s try:f(x) = xf(x) = x22

Graph it in pencil.Graph it in pencil.

Graph f(x-2) in a different Graph f(x-2) in a different color, but on the same color, but on the same

grid.grid.Graph f(x) – 4 in a Graph f(x) – 4 in a different color.different color.Graph -f(x) in a different Graph -f(x) in a different

color.color.Graph f(x + 3) + 1 in a Graph f(x + 3) + 1 in a different color.different color.

Let’s try:Let’s try:f(x) = |x|f(x) = |x|

Graph it in pencil.Graph it in pencil.

Graph f(x-1) in a different Graph f(x-1) in a different color, but on the same color, but on the same

grid.grid.Graph f(x) +2 in a Graph f(x) +2 in a different color.different color.Graph -f(-x) + 1 in a Graph -f(-x) + 1 in a different color.different color.Graph -f(x - 1) in a different Graph -f(x - 1) in a different

color.color.

Let’s try:Let’s try:

Graph it in pencil.Graph it in pencil.

Graph -f(x) in a different Graph -f(x) in a different color, but on the same color, but on the same

grid.grid.Graph f(-x) -1 in a Graph f(-x) -1 in a different color.different color.Graph f(x + 3) + 1 in a Graph f(x + 3) + 1 in a different color.different color.Graph -f(x) - 1 in a different Graph -f(x) - 1 in a different

color.color.

f(x) = f(x) = x x

Let’s try:Let’s try:

Graph it in pencil.Graph it in pencil.

Graph f(x - 1) + 3 in a Graph f(x - 1) + 3 in a different color, but on different color, but on

the same grid.the same grid.Graph -f(x + 2) in a Graph -f(x + 2) in a different color.different color.Graph f(x) -3 in a different Graph f(x) -3 in a different

color.color.Graph f(-x) in a different Graph f(-x) in a different color.color.

f(x) = f(x) = 3x – 1, x 3x – 1, x > 0 > 0 x + 1, x x + 1, x << 0 0

Non - Rigid Non - Rigid TransformationsTransformationsf(nx)f(nx)

nf(x)nf(x)

These distort the shape of These distort the shape of the graph.the graph.

Let’s see how Let’s see how non-rigid transformations work.non-rigid transformations work.

f(x) = |f(x) = |x|x|

2f(x2f(x))

f(2xf(2x))x y x y x y

Notice what happened to the y-Notice what happened to the y-value…value…

nf(x)nf(x)All the y’s of f(x) get All the y’s of f(x) get multipliedmultiplied by n. by n.

f(nx)f(nx)

All the x’s of f(x) getAll the x’s of f(x) get divideddivided by n.by n.

Let’s try:Let’s try:f(x) = xf(x) = x22

Graph it in pencil.Graph it in pencil.

Graph f(2x) in a different Graph f(2x) in a different color, but on the same color, but on the same

grid.grid.Graph 2f(x) in a different Graph 2f(x) in a different color.color.Graph Graph 11//22f(x) in a f(x) in a

different color.different color.Graph f(Graph f(11//22x) in a different x) in a different color.color.

Let’s create a unique Let’s create a unique shape and try one more shape and try one more

time.time.