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.) Compare the two parabolas.

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 -2 -1 1 2 x y -2 -1 1 2 4 6 1 3 2 1 3 4 6

Your Turn: Describe the transformation to the parent function: translated down 4 Describe the transformation to the parent function: translated up 5

Graphical Transformations Compare the two parabolas. Multiplying the parent function by 3, makes it 3 times as steep.

Why does multiplying the parent function by 3 cause the parent to be vertically stretched by a factor of 3?. Build a table of values for each equation for domain elements: -2, -1, 0, 1, 2. x y -2 -1 1 2 x y -2 -1 1 2 12 4 3 1 3 1 12 4

Graphical Transformations Multiplying the parent function by -1, reflects across the x-axis. Compare the two parabolas.

Your Turn: Describe the transformation to the parent function: Reflected across x-axis and translated up 2 Describe the transformation to the parent function: Vertically stretched by a factor of 3 and translated down 6

Graphical Transformations Compare the two parabolas.

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 -2 -1 1 2 x y -2 -1 1 2 9 4 4 1 1 1 1 4

Quadratic Transformations translating up or down Reflection across x-axis vertical stretch factor Translates left/right Reflected across x-axis, twice as steep, translated up 4, translated right 3

translated up 3 translated left 5 Your Turn: Describe the transformation to the parent function: translated up 3 translated left 5

Vertically stretched by a factor of 2, translated right 1 Your Turn: Describe the transformation to the parent function: Vertically stretched by a factor of 2, translated right 1

Your Turn: Describe the transformation to the parent function: Reflected across x-axis Vertically stretched by a factor of ½ (shrunk), translated up 4 translated left 3

Absolute Value Function Why does it have this shape?

Your turn: What is the transformation to the parent function? translated right 3 Vertically stretched by a factor of 2  Twice as steep Slope on right side is +2 slope on left side is -2

Reflected across x-axis Your turn: What is the transformation to the parent function? Reflected across x-axis VSF = 4  4 times as steep Left 2 up 4

Absolute Value Transformation Vertical stretch factor Translates left/right translating up or down Reflection across x-axis

Reflection across x-axis What does adding or subtraction “k” do to the parent function? Vertical shift What does adding or subtraction “h” do to the parent function? Horizontal shift What does multiplying by ‘a’ do to the parent function? Vertical stretch What does multiplying by (-1) do to the parent function? Reflection across x-axis

Square Root Function What is the domain of the graph?

Describe the transformation to the parent function: Up 4, right 2 Down 3, reflected across x-axis, VSF=2 left 3

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