Evaluating Functions

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 + 9 + 2x - 6 g(x-3) = x2 – 4x + 3 What does this mean?

  g(x) = x2 + 2x g(x-3) = x2 – 4x + 3 x = -b = 4 x = -b = -2 = 2 2a 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

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.

There are two kinds of transformations: Rigid Non-rigid

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

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.

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)

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

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.

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.

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.

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.

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

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…

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.

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.

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