The document discusses transformations of parent functions through various parameters. It introduces common parent functions and explains how changing parameters like a, h, and k transform the functions in different ways. The a parameter vertically stretches or shrinks functions, h shifts functions left or right along the x-axis, and k shifts functions up or down along the y-axis. Less common transformations include reflecting over the y-axis or horizontally stretching and shrinking using other parameters. Examples are provided to demonstrate combining multiple transformations on functions.

1. M2L6 TRANSFORMATIONS

2. Parent functions are the most basic form of a function.
They are centered or oriented around the origin (0,0).
See some of our most
common parent functions
below:
1. Linear 𝒚 = 𝒙
2. Quadratic 𝒚 = 𝒙 𝟐
3. Cubic 𝒚 = 𝒙 𝟑
4. Radical 𝒚 = 𝒙
5. Absolute value 𝒚 = 𝒙
6. Exponential 𝒚 = 𝟐 𝒙
7. Rational 𝒚 =
𝟏
𝒙

3. Putting numbers into the parent function transforms it into
something new. The letters a, h, & k are used to represent where
we place numbers.
1. Linear 𝒚 = 𝒂 𝒙 − 𝒉 + 𝒌
2. Quadratic 𝒚 = 𝒂(𝒙 − 𝒉) 𝟐 + 𝒌
3. Cubic 𝒚 = 𝒂(𝒙 − 𝒉) 𝟑 + 𝒌
4. Radical 𝒚 = 𝒂 𝒙 − 𝒉 + 𝒌
5. Absolute value 𝒚 = 𝒂 𝒙 − 𝒉 + 𝒌
6. Exponential 𝒚 = 𝒂(𝟐 𝒙−𝒉
) + 𝒌
7. Rational 𝒚 = 𝒂
𝟏
𝒙−𝒉
+ 𝒌 or 𝒚 =
𝒂
𝒙−𝒉
+ 𝒌

4. First, let’s look at what the acan do to a function.
If |a|>1, then it vertically stretches the function
(looks taller & skinnier)
If 0<|a|<1, then it vertically shrinks the function
(looks shorter and fatter)

5. -areflects the function over the x axis.

6. Next, let’s look at what the hcan do to a function.
If you have (x-h), the function
is shifted right h units
If you have (x+h), the function
is shifted left h units

7. Next, let’s look at what the kcan do to a function.
If you have (x)-k, the function
is shifted down k units
If you have (x)+k, the function
is shifted up k units

8. A less common transformation is reflecting the function over
the y axis (mirror image).
To create a y axis reflection, you negate the x inside
of it’s grouping symbols.
For example: 𝑦 = (−𝑥)2 OR 𝑦 = −𝑥 OR 𝑦 = 2−𝑥

9. Two more less common transformations are
horizontal stretching and horizontal shrinking.
To create a horizontal change, you
place a number in front of x inside of it’s grouping symbols.
For example: 𝑦 = (2𝑥)3
or 𝑦 =
1
2
𝑥 or 𝑦 = 11𝑥

10. Specifics on horizontal stretching
and horizontal shrinking:
• If the coefficient is >1, then it
horizontally shrinks the function
• If the coefficient is between 0
and 1, then it horizontally
stretches the function.

11. All of those transformation rules
work on any of our functions!
Let’s try an example. Given 𝑓 𝑥 = 𝑥3, shift it up 10 and left 7,
reflect it across the x axis, and vertically stretch it by 5.
• 𝑓 𝑥 = 𝑥3 + 10 now it’s shifted up 10
• 𝑓 𝑥 = (𝑥 + 7)3+10 now we’ve added the shift left 7
• 𝑓 𝑥 = −(𝑥 + 7)3 + 10 now we’ve reflected it over the x axis
• 𝑓 𝑥 = −5(𝑥 + 7)3 + 10 lastly, we vertically stretched it by 5

12. What if it already has transformations and we’re asked to change it?
Let’s try that. Given 𝑓 𝑥 = 7(2 𝑥) − 3, shift it up 4 and reflect it
across the y axis.
• 𝑓 𝑥 = 7 2 𝑥 − 3 + 4 this shifts it up 4, but we should combine
like terms
• 𝑓 𝑥 = 7 2 𝑥 + 1 now we just need to negate the x
• 𝑓 𝑥 = 7 2−𝑥
+ 1 It’s done. 

13. Here are some extra resources to help you:
Group of video tutorials:
• https://www.tes.com/lessons/x0Ei03pVKLMCxg/m2l6-
transformations-of-functions
Website with graph examples:
• http://www.regentsprep.org/regents/math/algtrig/atp9/funclesso
n1.htm
• https://www.mathsisfun.com/sets/function-transformations.html