Transformations of Linear Functions

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TRANSFORMATIONS
OF LINEAR
FUNCTIONS
The rules and what they mean:
y  f ( x ) This is our function
y  af ( x ) This is our function vertically stretched
1
y  f ( x ) This is our function vertically compressed
a
y  f (bx ) This is our function horizontally compressed
1 
y  f  x  This is our function horizontally stretched
b 
y   f ( x ) This is our function reflected over the x-axis
y  f ( x)
This is our function reflected over the y-axis
y  f ( x  c)
This is our function with a horizontal shift right
y  f ( x  c)
y  f ( x)  d
This is our function with a horizontal shift left
y  f ( x)  d
This is our function with a vertical shift down
This is our function with a vertical shift up
That’s a lot of rules… Now what?!
Let’s apply the rules to move
functions.
y = 3x Let’s start with this function
y = 3x shift the function horizontally right 3
horizontal movement will ALWAYS be inside with the x added
or subtracted and is OPPOSITE what you want.
To move the function horizontally, place the number
inside parenthesis and do the opposite of the way you
want to move. To move left put a plus and your number
and to move right put a minus and your number.
y = 3(x – 3)
Let’s try some more!
y = 3x horizontal shift left 4
y = 3(x + 4)
y = 3x horizontal shift right 5
y = 3(x - 5)
y = 3x horizontal shift left 7
y = 3(x + 7)
But what about up and down?
y = 3x shift the function vertically up 5
y = 3x + 5 just add on to the end.
Remember up you need a + to move up
and a – to move down. Vertical
movements do EXACTLY what they say.
y = 3x shift the function vertically down 2
y = 3x - 2
You try!
y = 3x vertical shift up 3
y = 3x + 3
y = 3x vertical shift down 8
y = 3x - 8
Put them together!
y = 3x
vertical shift down 5 and
horizontal shift right 6
y = 3(x – 6) – 5
Too easy? Let’s look at some others!
Vertically stretch y = 3x by a scale factor of 2
Simply put the 2 on the outside of 3x like this:
y = 2(3x)
That’s it???? Yep, that’s it! But what if it is a
compression? Same deal but you will see a
fraction.
Try it! Vertically compress y = 3x by a scale
factor of 1/4
1
y  3 x 
4
Horizontal compressions and stretches the
number will be inside touching the x.
If the number is a whole number it will
COMPRESS
If the number is a fraction it will STRETCH the
function.
y = 3x compress the function horizontally
by a scale factor of 2
y = 3(2x)
y = 3x stretch the function horizontally by a
scale factor of 1/2
1 
y  3 x 
2 
y = 3x reflect across the x-axis
y = -3x
y = 3x reflect across the y-axis
y = 3(-x)
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