Graphical Transformations Algebra-2 Graphical Transformations

Graphical Transformations Parent Function: The simplest function in a family of functions (lines, parabolas, cubic functions, etc.)

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

y = x y = x + 1 adding 1 to the parent function translates the graph up by 1. Compare the two lines. Each point moves up by one.

y = x y = x + 2 adding 2 to the parent function translates the graph up by 2. Compare the two lines. Each point moves up by two.

Your Turn: 1. Graph the parent function: y = x 2. Graph the line: 3. Explain how y = x – 3 transforms the parent function.

y = x y = 2x Compare the two lines. 1 1 1

The most simple line of all. y = x y = 2x The coefficient of ‘x’ makes the graph steeper. (vertically stretches the graph) Compare the two lines. 2 1

Your Turn: 4. Graph the following two lines on the same x-y axis. 1 5. Explain how the following equation transforms the parent function: y = 3x - 2. 2

Graphical Transformations Graph the lines. y = x Reflection across x-axis.

Graphical Transformations Transformation: an adjustment made to the parent function that results in a change to the graph of the parent function. translating up or down Reflection across x-axis more/less steep How does the graph of compare with the graph of ? Twice as steep, translated up 2

Your turn: 6. Explain how the following equation transforms the parent function: translated up 6 7. Explain how the following equation transforms the parent function: 5 times as steep, translated down 2 8. Explain how the following equation transforms the parent function: Reflected across x-axis, 3 times as steep, translated up 4

Translations to the left or right y = x Reflection across x-axis.

Translations to the left or right Describe the transformation 2 translated down 2 What if ‘x’ is replaced by (x + 3) translated left 3 -2

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

Graphical Transformations Parent Function: The simplest function in a family of functions (lines, parabolas, cubic functions, etc.) adding 2 to the parent function translates the graph up by 2. Compare the two parabolas. Each point moves up by two.

Your Turn: 9. Describe the transformation to the parent function: translated down 4 10. Describe the transformation to the parent function: translated up 5

Graphical Transformations Parent Function: The simplest function in a family of functions (lines, parabolas, cubic functions, etc.) Multiplying the parent function by 3, makes it 3 times as steep. Compare the two parabolas.

Graphical Transformations Parent Function: The simplest function in a family of functions (lines, parabolas, cubic functions, etc.) Multiplying the parent function by -1, reflects across the x-axis. Compare the two parabolas.

Your Turn: 11. Describe the transformation to the parent function: Reflected across x-axis and translated up 2 12. Describe the transformation to the parent function: 3 times as steep and translated down 6

Graphical Transformations Parent Function: The simplest function in a family of functions (lines, parabolas, cubic functions, etc.) Replacing ‘x’ with ‘x – 1’ translates the parent function right by 1. Compare the two parabolas.

Graphical Transformations Twice as steep, translated up 2 Reflection across x-axis more/less steep translating up or down Reflected across x-axis, twice as steep, translated up 4, translated right 3 Translates left/right

Your Turn: 13. Describe the transformation to the parent function: translated up 3 translated left 5 14. Describe the transformation to the parent function: 2 times as steep translated right 1 15. Describe the transformation to the parent function: Reflected across x-axis 1/2 as steep translated up 4 translated left 3

Absolute Value Function Clear your “y-editor” Enter the absolute value equation: “2nd” + “catalog” (“0” button) then “enter” Now graph it: Your turn: 1. What are the coordinates of the vertex? 2. What is the slope of the right side of the “vee”

Your turn: 16. Graph What is the transformation to the parent function? translated up 1 17. Clear from your “y-editor” then graph: What is the transformation to the parent function? translated right 1

Slope on right side is +2 slope on left side is -2 Your turn: 18. Clear from your “y-editor” then graph: What is the transformation to the parent function? Twice as steep Slope on right side is +2 slope on left side is -2

Your turn: 19. Clear from your “y-editor” then graph: What is the transformation to the parent function? Reflected across x-axis Slope on right side is -1 slope on left side is +1

Half as steep, translated down 5 reflected across x-axis more/less steep translating up or down Reflection across x-axis Three times as steep, translated down 2, translated left 5 Translates left/right 5 times as steep, translated up 3 translated right 2 reflected 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

Vocabulary Slope (h, k) rise run Vertex (h, k) Absolute Value Function: A function of the form: Slope (h, k) rise run Vertex (h, k)

Examples Slope of right side = ? vertex = ? Slope of right side = ?

Your Turn: Describe the transformation to the parent function. 20. 21. 22. 4 times as steep, translated: up 5, right 7 Reflected across x-axis, 2 times as steep, translated: up 2, left 3 Translated: down 6

Reflected across x-axis, 4 times as steep, translated: up 3 Your Turn: Describe the transformation to the parent function: 23. translated: up 6, right 5 24. Reflected across x-axis, 4 times as steep, translated: up 3 25. Reflected across x-axis, 2 times as steep, translated: down 0.25, right 19

Your turn: 26. The following graph is the function f(x). On the same axis graph the transformation: Points: (-2, -1) (-1, -1) ( 0, 0) ( 1, 1) ( 2, 1) ( 3, 0)