module 6.4 transforming linear functions
TRANSCRIPT
There are five ways you can transform a linear function:
1) Vertical Translation β Moving the entire graph (all points) up or down.2) Horizontal Translation β Moving the entire graph (all points) left or right.3) Stretch β The slope gets steeper4) Shrink β The slope gets less steep5) Reflection β The graph is reversed, like looking in a mirror
Module 6.4 β Transforming Linear Functions
A Family of Functions is a set of functions whose graphs have basic characteristics in common.
The most basic function within a Family of Functions is called the Parent Function.
ParentQuadratic Function
π = π π = ππ
ParentLinear Function
π = π π = π
ParentAbsolute Value Function
π = π π = |π|
Family ofAbsolute Value
Functions
Family ofQuadratic Functions
Family ofLinear Functions
Transformation #1:
Vertical Translation β Moving the entire graph (all points) up or down.
Do this by adding to or subtracting from the function. Here: ADDING TO
Notice that the y-intercept (b) got larger β by the amount that you added.
Parent
π = π π = π
π π = π + πSame slope,
different y-intercept
π π = π β π
Vertical Translation β By SUBTRACTING FROM the function.
Notice that the y-intercept (b) got smaller β by the amount that you subtracted.
Same slope,different y-intercept
Parent
π = π π = π
Transformation #2:
Horizontal Translation β Moving the entire graph (all points) left or right.
Do this by adding to or subtracting from the x itself.
BUT β WITH ALL HORIZONTAL TRANSLATIONS β THINK BACKWARDS
π β π means 2 to the RIGHT
π + π means 3 to the LEFT
Soβ¦
Moving a line DOWN 2 is the same as moving it to the RIGHT 2
Because it was π π = πBecause moving it DOWN 2 is π π = π β π (subtracting at the end)Moving it to the RIGHT 2 is π π = (π β π) (subtracting from the x)
Parent
π = π π = π
π π = π β ππ π = (π β π)
Transformation #3:
Stretch β The slope gets steeper. Think of a tightening rubber band.
Do this by multiplying the slope (m) by a number greater than 1..
π(π) = ππ
What happenswhen the multiplierbecomes very large?
π(π) = ππ
Parent
π(π) = π
Parent
π(π) = π
Transformation #4:
Shrink β The slope gets less steep. Think of a loosening rubber band.
Do this by multiplying the slope (m) by a number between 0 and 1..
π(π) =π
ππ
π¨π«π(π) = π. ππ
What happenswhen the multiplierbecomes very small?
π(π) =π
ππ
π¨π«π(π) = π. ππ
What happenswhen the multiplier
becomes less than 0?
Parent
π(π) = πParent
π(π) = π
Transformation #5:
Reflection β The graph is reversed, like looking in a mirror.
Do this by multiplying the slope (m) by β1..
π(π) = βπ
π(π) = βππ
Opposite slope
Parent
π(π) = π
π(π) = ππ
π π = ππ + π
The same 5 methods apply to Quadratic functions.Transformation #1: Vertical Translation βMoving the entire graph up or down.Do this by adding to or subtracting from the function. Here: ADDING TONotice that the vertex got larger βby the amount that you added.
π π = ππ β π
Parent
π = π π = ππ
Transformation #2:Horizontal Translation β Moving the entire graph left or right.Do this by adding to or subtracting from the x itself. BUT β WITH ALL HORIZONTAL TRANSLATIONS β THINK BACKWARDSπ β π means 2 to the RIGHTπ + π means 3 to the LEFT
Parent
π = π π = ππ
π π = (π β π)π
π π = (π + π)π
Transformation #3:
Stretch β The slope gets steeper.
Do this by multiplying the x-portionby a number greater than 1..
Transformation #4:
Shrink β The slope gets less steep.
Do this by multiplying the x-portionby a number between 0 and 1..
What happenswhen the multiplierbecomes very large?
Parent
π = π π = ππ
π π = πππ
π π =π
πππ
What happenswhen the multiplier
becomes less than 0?
What happenswhen the multiplierbecomes very small?
Transformation #5:
Reflection βThe graph is reversed, like looking in a mirror.
Do this by multiplying the x-portion by β1..
Parent
π = π π = ππ
π π = βππ
π π = |π| + π
The same 5 methods apply to Absolute Value functions.Transformation #1: Vertical Translation β Moving the entire graph up or down.Do this by adding to or subtracting from the function. Here: ADDING TONotice that the vertex got larger β by the amount that you added.
Parent
π = π π = |π|
Transformation #2:Horizontal Translation β Moving the entire graph left or right.Do this by adding to or subtracting from the x itself. BUT β WITH ALL HORIZONTAL TRANSLATIONS β THINK BACKWARDSπ β π means 2 to the RIGHTπ + π means 3 to the LEFT
Parent
π = π π = |π|
π π = |π β π|
Transformation #3:
Stretch β The slope gets steeper.
Do this by multiplying the absolute-value portionby a number greater than 1..
Transformation #4:
Shrink β The slope gets less steep.
Do this by multiplying the absolute-value portionby a number between 0 and 1..
What happenswhen the multiplierbecomes very large?
What happenswhen the multiplier
becomes less than 0?
What happenswhen the multiplierbecomes very small?
Parent
π = π π = |π|
π(π) = π|π|
π(π) = π. π|π|
Transformation #5:
Reflection βThe graph is reversed, like looking in a mirror.
Do this by multiplying the absolute-value portion by β1..
π(π) = β|π|
Parent
π = π π = |π|
π = π
π = ππ
π = |π|
Positive or NegativeSloping Up or Down Multiplier - Slope
Greater than 1 β StretchBetween 0 and 1 β Shrink
Positive or Negative NumberTranslation Up or Down
π = βππ + π
NegativeSloping Down
Multiplier - SlopeGreater than 1 β Stretch
Positive NumberTranslation Up
Positive or NegativeOpening Up or Down
MultiplierGreater than 1 β Stretch
Between 0 and 1 β Shrink
Positive or NegativeNumber β Translation
Up or Down
Positive or Negative NumberTranslation Left or Right
π = βΒ½ π β π π + π
NegativeOpening Down
MultiplierBetween 0 and 1 β Shrink Negative Number
Translation Right
Positive NumberTranslation Up
Positive or NegativeOpening Up or Down
MultiplierGreater than 1 β Stretch
Between 0 and 1 β Shrink
Positive or NegativeNumber β Translation
Up or Down
Positive or Negative NumberTranslation Left or Right
π = βπ π + π β π
NegativeOpening Down
MultiplierGreater than 1 β Stretch Positive Number
Translation Left
Negative NumberTranslation Down
You can combine transformations!For example, you can Translate it up,and then make it Stretch (steeper).
You can Translate it down,and then make it Shrink (less steep), and then Reflect it (mirror).
π(π) = π + π
π(π) = ππ + π
π(π) = βπ. ππ + π
π(π) = ππ + π
Now that you know how to change Parent functionsβ¦Change these functionsβ¦.so that they areβ¦
π π = π + π translated 3 units up
π π = ππ + π translated 3 units up and stretched by a factor of 2
π π = ππ + π translated 2 units down and shrunk by a factor of 3
π π =π
ππ + π reflected
π π = βππ + π translated 1 unit up and reflected
π π = βπ
ππ β π translated 3 units down and stretched by a factor 9
π π = βπ β π translated 6 units up, stretched by a factor 2, and reflected
A gym charges a one-time new member fee of $50 and then a monthly membership fee of $25.Graph it.What is the function?What is the slope? What then is the Rate-Of-Change?What is the y-intercept?
Say the gym increases the one-time fee to $60. Which aspect of the graph is changed? What happens to it?What is the new function?
Say the gym decreases the monthly charge to $20. Which aspect of the graph is changed? What happens to it?What is the new function?
Say the gym increases the one-time fee to $60 AND decreases the monthly charge to $20. What is the new function?
For large parties, a restaurant charges a reservation fee of $25, plus $15 per person. What is the total charge (function) for a party of x people?What is the charge for 50 people?
How will the graph of this function change if the reservation fee is raised to $50, and if the per-person charge is lowered to $12?
The number of chaperones on a field trip must include 1 teacher for every 4 students, plus a total of 2 parents. What is the function describing the number of chaperones for a trip of x students?How many chaperones are needed for 100 students?
How will the graph change if the number of parents is reduced to 0? How will the graph change if the number of teachers is raised to 1 for every 3 students?