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Essential Question: In the equation g(x) = c[a(x-b)] + d what do each of the letters do to the graph?

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3.4: Graphs and Transformations Parent function: A function with a certain shape that has the simplest rule for that shape. For example, f(x) = x 2 is the simplest rule for a parabola Any parabola is a transformation of that parent function All of the following parent functions are on page 173 in your books… there is no need to copy them now. It’s most important that you get down the words in blue. Everything else is predominately mathematical definition.

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3.4: Graphs and Transformations Identify the parent function.

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3.4: Graphs and Transformations Constant functionIdentity Function f(x) = 1f(x) = x

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3.4: Graphs and Transformations Absolute-value functionGreatest Integer Function f(x) = |x|f(x) = [x]

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3.4: Graphs and Transformations Quadratic functionCubic Function f(x) = x 2 f(x) = x 3

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3.4: Graphs and Transformations Reciprocal function Square Root Function f(x) = 1/xf(x) =

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3.4: Graphs and Transformations Cube root function f(x) =.

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3.4: Graphs and Transformations Vertical shifts When a value is added to f(x), the effect is to add the value to the y-coordinate of each point, effectively shifting the graph up and down. If c is a positive number, then: The graph g(x) = f(x) + c is the graph of f shifted up c units The graph g(x) = f(x) – c is the graph of f shifted down c units Numbers adjusted after the parent function affect the graph vertically, as one would expect (+ up, – down)

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3.4: Graphs and Transformations Horizontal shifts When a value is added to the x of a function, the effect is to readjust the graph, effectively shifting the graph left and right. If c is a positive number, then: The graph g(x) = f(x+c) is the graph of f shifted c units to the left The graph g(x) = f(x-c) is the graph of f shifted c units to the right Numbers adjusted to the x of the parent function [inside a parenthesis] affect the graph horizontally, in reverse of expected values (+ left, – right)

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3.4: Graphs and Transformations Reflections Adding a negative sign before a function reflects the graph about the x-axis. Adding a negative sign before the x in the function reflects the graph about the y-axis. A negative sign before the function flips up & down (vertically, also called “reflected across the x-axis”) A negative sign before the x flips left & right (horizontally, also called “reflected across the y-axis”)

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3.4: Graphs and Transformations Stretches & Compressions (Vertical) If a function is multiplied by a number, it will stretch or compress the parent function vertically If c > 1, then the graph g(x) = c f(x) is the graph of f stretched vertically (away from the x-axis) by a factor of c If c < 1, then the graph g(x) = c f(x) is the graph of f compressed vertically (towards the x-axis) by a factor of c Multiplying the entire function will stretch or compress a function (proportionally) towards or away from the x-axis, as expected (large numbers stretch, small numbers compress)

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3.4: Graphs and Transformations Stretches & Compressions (Horizontal) If the x of a function is multiplied by a number, it will stretch or compress the parent function horizontally If c > 1, then the graph g(x) = f(c x) is the graph of f compressed horizontally (towards the y-axis) by a factor of 1/c If c < 1, then the graph g(x) = f(c x) is the graph of f stretched horizontally (away from the y-axis) by a factor of 1/c Multiplying the x of a function will stretch or compress a function (inversely) away from or towards the y-axis, opposite as expected (large numbers compress by the reciprocal, small numbers stretch by the reciprocal)

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3.4: Graphs and Functions Assignment Page 182 1-21, odd problems

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Essential Question: In the equation g(x) = c[a(x-b)] + d what do each of the letters do to the graph?

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3.4: Graphs and Transformations We have our grand equation: g(x) = c[a(x-b)] + d Addition on the outside shifts the graph vertically (d) (as expected: positive == up, negative == down) A negative on the outside flips the function vertically (c) Multiplication on the outside stretches/compresses the graph vertically (c) (as expected: large numbers == stretch, small numbers == compress) Addition on the inside of the parenthesis shifts the graph horizontally (b) (opposite as expected: positive == left, negative == right) A negative on the inside of the parenthesis flips the graph horizontally (a) Multiplication on the inside stretches/compresses the graph horizontally (a) (opposite: large numbers == compress by inverse, small numbers == stretch by inverse)

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3.4: Graphs and Transformations We have our grand equation: g(x) = c[a(x-b)] + d Order of application a (horizontal reflection) a (horizontal stretch/compression) b (horizontal shift) c (vertical reflection) c (vertical stretch/compression) d (vertical shift)

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3.4: Graphs and Transformations Example set #1 – write a rule for the function whose graph can be obtained from the given parent function by performing the given transformation. Parent function: f(x) = x 2 Transformations: shift 5 units left and up 4 units Parent function: f(x) = Transformations: shift 2 units right, stretched horizontally by a factor of 2, and shift up 2 units g(x) = (x + 5) 2 + 4

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3.4: Graphs and Transformations Example set #2 – describe a sequence of transformations that transform the graph of the parent function f into the graph of the function g. Do not graph the function. a = -1/2, b = 6 (b is always its opposite), c = -1, d = 0 1)Horizontal reflection 2)Horizontal stretch by a factor of 2 (horizontal stretches are inverses) 3)Horizontal shift right 6 units 4)Vertical reflection

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3.4: Graphs and Transformations Example set #2 – describe a sequence of transformations that transform the graph of the parent function f into the graph of the function g. Do not graph the function. a = -2, b = 2, c = 3, d = 0 1)Horizontal reflection 2)Horizontal compression by a factor of 0.5 3)Horizontal shift right 2 units 4)Vertical stretch by a factor of 3

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3.4: Graphs and Functions Assignment Page 182 23-41, odd problems

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