SECONDARY MATH 3 3-3 TRANSFORMATIONS
WARM UP
WHAT YOU WILL LEARN How to transform functions from the parent function How to describe a transformation How to write an equation of a transformed function
There are several tpes of functions (linear, quadratic, absolute value, sine, cosine, eponential, cubic, etc.). Each of these can be considered a famil with unique characteristics that are shared among the members. The parent function is the most basic function in each famil. It is used to create more complicated functions.
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, Shifting (translating) the graph left or right vertical stretching or shrinking (making the graph more steep or less steep) Reflecting across -ais or -ais
GRAPHICAL TRANSFORMATIONS = = + Compare the two parabolas. Adding or Subtracting a number from the parent function moves the graph up or down.
= Wh does adding to the parent function translate the graph up b? = + Build a table of values for each equation for domain elements: -, -, 0,,. - - 0 4 0 4 - - 0 6 3 3 6
YOUR TURN: Describe the transformation to the parent function: = = 4 translated down 4 = + 5 translated up 5 What ke feature is affected b a vertical shift? Range
GRAPHICAL TRANSFORMATIONS = = 3 Compare the two parabolas. Multipling the -values of the parent function b 3, makes it 3 times as steep.
= Wh does multipling the parent function b 3 cause the parent to be verticall stretched b a factor of 3?. = 3 Build a table of values for each equation for domain elements: -, -, 0,,. - - 0 4 0 4 What values are affected b a vertical stretch? The values. You multipl the values b the stretch factor. - - 0 3 0 3
GRAPHICAL TRANSFORMATIONS = = Multipling the parent function b -, reflects across the -ais. Compare the two parabolas.
YOUR TURN: Describe the transformation to the parent function: = = + Reflected across -ais and translated up = 3 6 Verticall stretched b a factor of 3 and translated down 6 What ke feature is affected b a reflection across the -ais? Maimum or minimum
GRAPHICAL TRANSFORMATIONS = = ( ) Compare the two parabolas. Adding or Subtracting a number to the variable of the parent function moves the graph left right. What ke feature ma be affected b a horizontal shift? Domain
= Wh does replacing with translates the parent function right b. = ( ) Build a table of values for each equation for domain elements: -, -, 0,,. - - 0 4 0 4 - - 0 9 4 0
QUADRATIC TRANSFORMATIONS ) = ( ) a( h + k Reflection across -ais vertical stretch factor Translates left/right translating up or down = ( 3) + 4 Reflected across -ais, twice as steep, translated up 4, translated right 3
DESCRIBE THE TRANSFORMATION FROM THE PARENT FUNCTION AND THEN WRITE THE EQUATION FOR THE FOLLOWING GRAPHS. translated up 3 translated left 5 = ( + 5) + 3 Verticall stretched b a factor of, translated right = ( )
YOUR TURN: Describe the transformation to the parent function: = = ( + 3) + 4 Reflected across -ais Verticall stretched b a factor of ½ (shrunk), translated up 4 translated left 3
Absolute Value Transformation = ( ) a h + k Reflection across -ais Vertical stretch factor Translates left/right translating up or down
YOUR TURN: = What is the transformation to the parent function? = 3 translated right 3 = Verticall stretched b a factor of Twice as steep Slope on right side is + slope on left side is -
YOUR TURN: = What is the transformation to the parent function? = 3 + + 4 Reflected across -ais Vertical stretch b a factor of 3 times as steep Left up 4
Square Root Transformation = ( ) a h + k Reflection across -ais Vertical stretch factor Translates left/right translating up or down
= 4 + Up 4, right = + 4 Notice that both equations are the same equation. = 3 + 3 Down 3, reflected across -ais, Vertical stretch b a factor of left 3
What does adding or subtraction k do to the parent function? f ( ) = + k Vertical shift What does adding or subtraction h do to the parent function? f ( ) = h What does multipling b a do to the parent function? f ( ) = a Horizontal shift Vertical stretch What does multipling b (-) do to the parent function? f ( ) = Reflection across -ais