Module 6.4

Transforming Linear Functions

What are the ways in which you can transform the graph of a linear function?

P. 269

There are 4 ways you can transform a linear function:

1) Translation – Moving the entire graph (all points) up or down.2) Stretch – The slope gets steeper3) Shrink – The slope gets less steep4) Reflection – The graph is reversed, like looking in a mirror

Transformation #1:

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.

𝒇 𝒙 = 𝒙

𝒇 𝒙 = 𝒙 + 𝟐Same slope,

different y-intercept

𝒇 𝒙 = 𝒙

𝒇 𝒙 = 𝒙 − 𝟐

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

Transformation #2:

Stretch – The slope gets steeper.

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

𝒇(𝒙) = 𝒙

𝒈(𝒙) = 𝟐𝒙

What happens when the multiplier becomes very large?

𝒇(𝒙) = 𝒙

𝒈(𝒙) = 𝟑𝒙

Transformation #3:

Shrink – The slope gets less steep.

Do this by multiplying the slope (m) by a number between 0 and 1..

𝒇(𝒙) = 𝒙

𝒈(𝒙) =𝟏

𝟐𝒙

𝐨𝐫𝒈(𝒙) = 𝟎. 𝟓𝒙

𝒇(𝒙) = 𝒙

What happens when the multiplier becomes very small?

𝒈(𝒙) =𝟏

𝟓𝒙

𝐨𝐫𝒈(𝒙) = 𝟎. 𝟐𝒙

Transformation #4:

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

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

𝒇(𝒙) = 𝒙𝒈(𝒙) = −𝒙

𝒇(𝒙) = 𝟑𝒙𝒈(𝒙) = −𝟑𝒙

Opposite slope

The same 4 methods apply to absolute value functions.

1) Translation – Moving the entire graph (all points) up or down.2) Stretch – The slope gets steeper3) Shrink – The slope gets less steep4) Reflection – The graph is reversed, like looking in a mirror

𝒇 𝒙 = |𝒙|

𝒇 𝒙 = |𝒙| + 𝟐

Transformation #1:

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.

Transformation #2:

Stretch – The slope gets steeper.

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

𝒇(𝒙) = |𝒙|

𝒈(𝒙) = 𝟐|𝒙|

What happens when the multiplier becomes very large?

Transformation #3:

Shrink – The slope gets less steep.

Do this by multiplying the slope (m) by a number between 0 and 1..

𝒇(𝒙) = |𝒙|

𝒈(𝒙) = 𝟎. 𝟒|𝒙|What happens when the multiplier becomes very small?

Transformation #4:

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

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

𝒇(𝒙) = |𝒙|

𝒈(𝒙) = −|𝒙|

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

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.

What is the function?What is the slope?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?

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 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?