There are five ways you can transform a linear function:

1) Vertical Translation – Moving the entire graph (all points) up or down.2) Horizontal Translation – Moving the entire graph (all points) left or right.3) Stretch – The slope gets steeper4) Shrink – The slope gets less steep5) Reflection – The graph is reversed, like looking in a mirror

Module 6.4 – 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

𝒚 = 𝒇 𝒙 = 𝒙𝟐

ParentLinear Function

𝒚 = 𝒇 𝒙 = 𝒙

ParentAbsolute Value Function

𝒚 = 𝒇 𝒙 = |𝒙|

Family ofAbsolute Value

Functions

Family ofQuadratic Functions

Family ofLinear Functions

Transformation #1:

Vertical Translation – 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 – by the amount that you added.

Parent

𝒚 = 𝒇 𝒙 = 𝒙

𝒈 𝒙 = 𝒙 + 𝟐Same slope,

different y-intercept

𝒈 𝒙 = 𝒙 − 𝟐

Vertical Translation – By SUBTRACTING FROM the function.

Notice that the y-intercept (b) got smaller – by the amount that you subtracted.

Same slope,different y-intercept

Parent

𝒚 = 𝒇 𝒙 = 𝒙

Transformation #2:

Horizontal Translation – Moving the entire graph (all points) left or right.

Do this by adding to or subtracting from the x itself.

BUT – WITH ALL HORIZONTAL TRANSLATIONS – THINK BACKWARDS

𝒙 − 𝟐 means 2 to the RIGHT

𝒙 + 𝟑 means 3 to the LEFT

So…

Moving a line DOWN 2 is the same as moving it to the RIGHT 2

Because it was 𝒇 𝒙 = 𝒙Because moving it DOWN 2 is 𝒈 𝒙 = 𝒙 − 𝟐 (subtracting at the end)Moving it to the RIGHT 2 is 𝒈 𝒙 = (𝒙 − 𝟐) (subtracting from the x)

Parent

𝒚 = 𝒇 𝒙 = 𝒙

𝒈 𝒙 = 𝒙 − 𝟐𝒈 𝒙 = (𝒙 − 𝟐)

Transformation #3:

Stretch – The slope gets steeper. Think of a tightening rubber band.

Do this by multiplying the slope (m) by a number greater than 1..

𝒈(𝒙) = 𝟐𝒙

What happenswhen the multiplierbecomes very large?

𝒈(𝒙) = 𝟑𝒙

Parent

𝒇(𝒙) = 𝒙

Parent

𝒇(𝒙) = 𝒙

Transformation #4:

Shrink – 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..

𝒈(𝒙) =𝟏

𝟐𝒙

𝐨𝐫𝒈(𝒙) = 𝟎. 𝟓𝒙

What happenswhen the multiplierbecomes very small?

𝒈(𝒙) =𝟏

𝟓𝒙

𝐨𝐫𝒈(𝒙) = 𝟎. 𝟐𝒙

What happenswhen the multiplier

becomes less than 0?

Parent

𝒇(𝒙) = 𝒙Parent

𝒇(𝒙) = 𝒙

Transformation #5:

Reflection – The graph is reversed, like looking in a mirror.

Do this by multiplying the slope (m) by –1..

𝒈(𝒙) = −𝒙

𝒈(𝒙) = −𝟑𝒙

Opposite slope

Parent

𝒇(𝒙) = 𝒙

𝒇(𝒙) = 𝟑𝒙

𝒈 𝒙 = 𝒙𝟐 + 𝟐

The same 5 methods apply to Quadratic functions.Transformation #1: Vertical Translation –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 –by the amount that you added.

𝒉 𝒙 = 𝒙𝟐 − 𝟑

Parent

𝒚 = 𝒇 𝒙 = 𝒙𝟐

Transformation #2:Horizontal Translation – Moving the entire graph left or right.Do this by adding to or subtracting from the x itself. BUT – WITH ALL HORIZONTAL TRANSLATIONS – THINK BACKWARDS𝒙 − 𝟐 means 2 to the RIGHT𝒙 + 𝟑 means 3 to the LEFT

Parent

𝒚 = 𝒇 𝒙 = 𝒙𝟐

𝒈 𝒙 = (𝒙 − 𝟐)𝟐

𝒉 𝒙 = (𝒙 + 𝟑)𝟐

Transformation #3:

Stretch – The slope gets steeper.

Do this by multiplying the x-portionby a number greater than 1..

Transformation #4:

Shrink – 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

𝒚 = 𝒇 𝒙 = 𝒙𝟐

𝒈 𝒙 = 𝟑𝒙𝟐

𝒈 𝒙 =𝟏

𝟑𝒙𝟐

What happenswhen the multiplier

becomes less than 0?

What happenswhen the multiplierbecomes very small?

Transformation #5:

Reflection –The graph is reversed, like looking in a mirror.

Do this by multiplying the x-portion by –1..

Parent

𝒚 = 𝒇 𝒙 = 𝒙𝟐

𝒈 𝒙 = −𝒙𝟐

𝒇 𝒙 = |𝒙| + 𝟐

The same 5 methods apply to Absolute Value functions.Transformation #1: Vertical Translation – 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 – by the amount that you added.

Parent

𝒚 = 𝒇 𝒙 = |𝒙|

Transformation #2:Horizontal Translation – Moving the entire graph left or right.Do this by adding to or subtracting from the x itself. BUT – WITH ALL HORIZONTAL TRANSLATIONS – THINK BACKWARDS𝒙 − 𝟐 means 2 to the RIGHT𝒙 + 𝟑 means 3 to the LEFT

Parent

𝒚 = 𝒇 𝒙 = |𝒙|

𝒈 𝒙 = |𝒙 − 𝟑|

Transformation #3:

Stretch – The slope gets steeper.

Do this by multiplying the absolute-value portionby a number greater than 1..

Transformation #4:

Shrink – 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

𝒚 = 𝒇 𝒙 = |𝒙|

𝒈(𝒙) = 𝟐|𝒙|

𝒈(𝒙) = 𝟎. 𝟒|𝒙|

Transformation #5:

Reflection –The graph is reversed, like looking in a mirror.

Do this by multiplying the absolute-value portion by –1..

𝒈(𝒙) = −|𝒙|

Parent

𝒚 = 𝒇 𝒙 = |𝒙|

𝒚 = 𝒙

𝒚 = 𝒙𝟐

𝒚 = |𝒙|

Positive or NegativeSloping Up or Down Multiplier - Slope

Greater than 1 – StretchBetween 0 and 1 – Shrink

Positive or Negative NumberTranslation Up or Down

𝒚 = −𝟑𝒙 + 𝟐

NegativeSloping Down

Multiplier - SlopeGreater than 1 – Stretch

Positive NumberTranslation Up

Positive or NegativeOpening Up or Down

MultiplierGreater than 1 – Stretch

Between 0 and 1 – Shrink

Positive or NegativeNumber – Translation

Up or Down

Positive or Negative NumberTranslation Left or Right

𝒚 = −½ 𝒙 − 𝟏 𝟐 + 𝟐

NegativeOpening Down

MultiplierBetween 0 and 1 – Shrink Negative Number

Translation Right

Positive NumberTranslation Up

Positive or NegativeOpening Up or Down

MultiplierGreater than 1 – Stretch

Between 0 and 1 – Shrink

Positive or NegativeNumber – Translation

Up or Down

Positive or Negative NumberTranslation Left or Right

𝒚 = −𝟑 𝒙 + 𝟏 − 𝟐

NegativeOpening Down

MultiplierGreater than 1 – 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).

𝒇(𝒙) = 𝒙 + 𝟏

𝒈(𝒙) = 𝟐𝒙 + 𝟐

𝒈(𝒙) = −𝟎. 𝟓𝒙 + 𝟏

𝒇(𝒙) = 𝟐𝒙 + 𝟐

Now that you know how to change Parent functions…Change these functions….so that they are…

𝒇 𝒙 = 𝒙 + 𝟏 translated 3 units up

𝒇 𝒙 = 𝟒𝒙 + 𝟔 translated 3 units up and stretched by a factor of 2

𝒇 𝒙 = 𝟔𝒙 + 𝟏 translated 2 units down and shrunk by a factor of 3

𝒇 𝒙 =𝟏

𝟐𝒙 + 𝟏 reflected

𝒇 𝒙 = −𝟖𝒙 + 𝟏 translated 1 unit up and reflected

𝒇 𝒙 = −𝟏

𝟑𝒙 − 𝟕 translated 3 units down and stretched by a factor 9

𝒇 𝒙 = −𝒙 − 𝟐 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?