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r 2 3r 54 2x2 4x 4 Lesson 1-6 Graphical Transformations Graphical Transformations Transformation: an adjustment made to the parent function that results in a change to the graph of the parent function. Changes could include: shifting (“translating”) the graph up or down, “translating” the graph left or right vertical stretching or shrinking (making the graph more steep or less steep) Reflecting across x-axis or y-axis Graphical Transformations Parent Function: The simplest function in a family of functions (lines, parabolas, cubic functions, etc.) yx 2 y x 2 2 Compare the two parabolas. yx y x 2 2 2 Why does adding 2 to the parent function shift the graph up by 2? Build a table of values for each equation for domain elements: -2, -1, 0, 1, 2. x y x y -2 4 1 0 1 4 -2 6 3 2 3 6 -1 0 1 2 -1 0 1 2 Your Turn: Describe the transformation to the parent function: y x 4 2 translated down 4 Describe the transformation to the parent function: y x 5 2 yx 2 translated up 5 yx 2 Graphical Transformations yx y 3x 2 Compare the two parabolas. Multiplying the parent function by 3, makes it 3 times as steep. 2 yx 2 Why does multiplying the parent function by 3 cause the parent to be vertically stretched by a factor of 3?. y 3x 2 Build a table of values for each equation for domain elements: -2, -1, 0, 1, 2. x -2 -1 0 1 2 y 4 1 0 1 4 x y -2 12 3 0 3 12 -1 0 1 2 Graphical Transformations yx 2 y x2 Multiplying the parent function by -1, reflects across the x-axis. Compare the two parabolas. Your Turn: Describe the transformation to the parent function: y x 2 2 Reflected across x-axis and translated up 2 Describe the transformation to the parent function: y 3x 6 2 y x2 Vertically stretched by a factor of 3 and translated down 6 yx 2 Graphical Transformations yx 2 y ( x 1) Compare the two parabolas. 2 yx y ( x 1) 2 Why does replacing ‘x’ with ‘x – 1’ translates the parent function right by 1. Build a table of values for each equation for domain elements: -2, -1, 0, 1, 2. x y x y -2 9 -2 4 -1 4 -1 1 0 1 0 0 1 0 1 1 2 1 2 4 2 Quadratic Transformations y (1)a( x h) k 2 Reflection across x-axis vertical stretch factor Translates left/right translating up or down y 2( x 3) 4 2 Reflected across x-axis, twice as steep, translated up 4, translated right 3 Your Turn: Describe the transformation to the parent function: y ( x 5) 3 2 translated up 3 translated left 5 y x2 Your Turn: Describe the transformation to the parent function: 2 y 2( x 1) Vertically stretched by a factor of 2, translated right 1 y x2 Your Turn: Describe the transformation to the parent function: 1 2 y ( x 3) 4 2 Reflected across x-axis Vertically stretched by a factor of ½ (shrunk), translated up 4 translated left 3 y x2 Absolute Value Function Why does it have this shape? f ( x) x Your turn: y x What is the transformation to the parent function? y x 3 translated right 3 y 2x Vertically stretched by a factor of 2 Twice as steep Slope on right side is +2 slope on left side is -2 Your turn: y x What is the transformation to the parent function? y 3 x 2 4 Reflected across x-axis VSF = 4 4 times as steep Left 2 up 4 Absolute Value Transformation y (1)a x h k Reflection across x-axis Vertical stretch factor Translates left/right translating up or down What does adding or subtraction “k” do to the parent function? f ( x) x k Vertical shift What does adding or subtraction “h” do to the parent function? f ( x) x h Horizontal shift What does multiplying by ‘a’ do to the parent function? f ( x) a x Vertical stretch What does multiplying by (-1) do to the parent function? f ( x) x Reflection across x-axis Square Root Function What is the domain of the graph? f ( x) x Describe the transformation to the parent function: y 4 x2 y 3 2 x 3 Up 4, right 2 y x2 4 Down 3, reflected across x-axis, VSF=2 left 3 y (1)a x h k Reflecting Across the x-axis Reflecting across the y-axis example Question: • What happens when an even function is reflected across the y-axis? Homework: • HW 1-6 pg 147: 2-32 Even