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Published byMackenzie Donahue Modified over 2 years ago

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

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g(x) = x 2 + 2x g(x-3) = x 2 – 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

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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.

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There are two kinds of transformations: Rigid Non-rigid

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

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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. repositioned.

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f(x) + c Moves up c squares. (Adding c to all ys) f(x) - c Moves down c squares. (Subtracting c from all ys) f(x + c) Moves left c squares. (Subtracting c from all xs) f(x - c) Moves right c squares. (Adding c to all xs) all xs)

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

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Lets try: f(x) = x 2 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.

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Lets 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.

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Lets try: 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. f(x) = x

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Lets try: 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. f(x) = 3x – 1, x > 0 x + 1, x < 0

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

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Lets see how non-rigid transformations work. f(x) = |x| 2f(x) f(2x) xy xyxy Notice what happened to the y-value…

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nf(x) All the ys of f(x) get multiplied by n. f(nx) All the xs of f(x) get divided by n.

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Lets try: f(x) = x 2 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 / 2 f(x) in a different color. Graph f( 1 / 2 x) in a different color.

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Lets create a unique shape and try one more time.

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