1. Mrs. Rivas 𝒚=𝒂 𝒙−𝒉 +𝒌 𝒚=𝟒 𝒙−𝟑 +(−𝟓) 𝒚=𝟒 𝒙−𝟑 −𝟓 𝒚=𝟒𝒙−𝟏𝟐−𝟓 𝒚=𝟒𝒙−𝟏𝟕
Write an equation of a line in slope intercept form given the
point and the slope of the line. Use 𝒚=𝒂 𝒙−𝒉 +𝒌.
a = 4 and point (3, -5)
(𝒉, 𝒌)
𝒚=𝒂 𝒙−𝒉 +𝒌
𝒚=𝟒 𝒙−𝟑 +(−𝟓)
𝒚=𝟒 𝒙−𝟑 −𝟓
Distributive Prop.
𝒚=𝟒𝒙−𝟏𝟐−𝟓
𝒚=𝟒𝒙−𝟏𝟕

2. Mrs. Rivas Point slope form 𝒚− 𝒚 𝟏 =𝒎(𝒙− 𝒙 𝟏 ) + 𝒚 𝟏 + 𝒚 𝟏
Ida S. Baker H.S.
Point slope form
𝒚− 𝒚 𝟏 =𝒎(𝒙− 𝒙 𝟏 )
+ 𝒚 𝟏
+ 𝒚 𝟏
𝒚=𝒎(𝒙− 𝒙 𝟏 )+ 𝒚 𝟏
𝒙, 𝒚 =(𝒉,𝒌) 𝒂 𝒉 𝒌
𝑵𝒆𝒘
𝒚=𝒂(𝒙−𝒉)+𝒌

3. Mrs. Rivas Example: 𝒚=𝒂 𝒙−𝒉 +𝒌 𝒚=𝟔 𝒙−(−𝟑) +𝟓 𝒚=𝟔 𝒙+𝟑 +𝟓 𝒚=𝟔𝒙+𝟏𝟖+𝟓
Ida S. Baker H.S.
Example:
Write an equation of a line in slope intercept form given the
point and the slope of the line. Use 𝒚=𝒂 𝒙−𝒉 +𝒌.
a = 6 and point (-3, 5)
(𝒉, 𝒌)
𝒚=𝒂 𝒙−𝒉 +𝒌
𝒚=𝟔 𝒙−(−𝟑) +𝟓
𝒚=𝟔 𝒙+𝟑 +𝟓
Distributive Prop.
𝒚=𝟔𝒙+𝟏𝟖+𝟓
𝒚=𝟔𝒙+𝟐𝟑

4. Mrs. Rivas Do it: 𝒚=𝒂 𝒙−𝒉 +𝒌 𝒚= 𝟏 𝟓 𝒙−𝟓 +𝟎 𝒚= 𝟏 𝟓 𝒙−𝟏+𝟎 𝒚= 𝟏 𝟓 𝒙 −𝟏
Ida S. Baker H.S.
Do it:
Write an equation of a line in slope intercept form given the
point and the slope of the line. Use 𝒚=𝒂 𝒙−𝒉 +𝒌.
a = 𝟏 𝟓 and point (5, 0)
(𝒉, 𝒌)
𝒚=𝒂 𝒙−𝒉 +𝒌
𝒚= 𝟏 𝟓 𝒙−𝟓 +𝟎
Distributive Prop.
𝒚= 𝟏 𝟓 𝒙−𝟏+𝟎
𝒚= 𝟏 𝟓 𝒙 −𝟏

5. International Studies Charter School.
Mrs. Rivas
International Studies Charter School.
Section 2-6
Families of Functions
Objective: To analyze transformations of functions.
Vocabulary:
Parent Function: is the simplest form in a set of functions that form a family.
Example:
𝒇 𝒙 =𝒙 is the parent function of 𝒇 𝒙 =𝟑 𝒙+𝟐 −𝟏
𝒇 𝒙 = 𝒙 is the parent function of 𝒇 𝒙 =𝟑 𝒙+𝟐 −𝟏
𝒇 𝒙 = 𝒙 𝟐 is the parent function of 𝒇 𝒙 =𝟑 𝒙+𝟐 𝟐 −𝟏

6. International Studies Charter School.
Mrs. Rivas
International Studies Charter School.
Section 2-6
Families of Functions
Objective: To analyze transformations of functions.
Vocabulary:
Transformation: a transformation of a function
𝑓(𝑥) = 𝑎(𝑥−ℎ)+𝑘 is a change made to at least one of the values a, h, and k. The four type of transformations are dilations, reflections, rotation, and translations.
Example:
𝒈 𝒙 =𝟐 𝒙−𝟑 𝟐 is a transformation of𝒇 𝒙 = 𝒙 𝟐

7. International Studies Charter School.
Mrs. Rivas
International Studies Charter School.
Section 2-6
Families of Functions
Objective: To analyze transformations of functions.
Vocabulary:
Translation: it shifts the graph of the parent function horizontally, vertically, or both without changing its shape or orientation.
Example:
𝒈 𝒙 =𝟐 𝒙−𝟑 𝟐 is a transformation of𝒇 𝒙 = 𝒙 𝟐

8. International Studies Charter School.
Mrs. Rivas
International Studies Charter School.
Section 2-6
Families of Functions

9. International Studies Charter School.
Mrs. Rivas
International Studies Charter School.
Section 2-6
Families of Functions

10. International Studies Charter School.
Mrs. Rivas
International Studies Charter School.
Section 2-6
Families of Functions

11. International Studies Charter School.
Mrs. Rivas
International Studies Charter School.
Section 2-6
Families of Functions

12. International Studies Charter School.
Mrs. Rivas
International Studies Charter School.
Section 2-6
Families of Functions

13. International Studies Charter School.
Mrs. Rivas
International Studies Charter School.
Section 2-6
Families of Functions
Vocabulary:
Reflection: flips the graph of a function across a line, such as the x- or y-axis. Each point on the graph of the reflected function is the same distance from the line of the reflection as its corresponding point on the graph of the original function.
Example:

14. Mrs. Rivas Families of Functions 𝒈 𝒙 =𝒇(−𝒙) =𝟑 −𝒙 +𝟑 𝒈 𝒙 =−𝟑 𝒙 +𝟑
International Studies Charter School.
Section 2-6
Families of Functions
A reflection in the y-axis change the sign of x.
𝒈 𝒙 =𝒇(−𝒙)
=𝟑 −𝒙 +𝟑
Evaluate f(-x) and simplify
𝒈 𝒙 =−𝟑 𝒙 +𝟑
You can check by graphing

15. International Studies Charter School.
Mrs. Rivas
International Studies Charter School.
Section 2-6
Families of Functions

16. International Studies Charter School.
Mrs. Rivas
International Studies Charter School.
Section 2-6
Families of Functions
𝒚=𝒂 𝒙−𝒉 𝟐 +𝒌
Vocabulary:
Vertical Stretch:
when 𝒂 >𝟏.
Vertical compression:
when 0<𝒂<𝟏.
Example: 𝒚=𝟑 𝒙+𝟐 𝟐 −𝟏
Example: 𝒚= 𝟏 𝟑 𝒙+𝟐 𝟐 −𝟏

17. International Studies Charter School.
Mrs. Rivas
International Studies Charter School.
Section 2-6
Families of Functions

18. International Studies Charter School.
Mrs. Rivas
International Studies Charter School.
Section 2-6
Families of Functions

19. International Studies Charter School.
Mrs. Rivas
International Studies Charter School.
Section 2-6
Families of Functions

20. International Studies Charter School.
Mrs. Rivas
International Studies Charter School.
Section 2-6
Families of Functions

21. International Studies Charter School.
Mrs. Rivas
International Studies Charter School.
Section 2-6
Families of Functions

22. International Studies Charter School.
Mrs. Rivas
International Studies Charter School.
Section 2-7
Absolute value Functions and graphs

23. International Studies Charter School.
Mrs. Rivas
International Studies Charter School.
Section 2-7
Absolute value Functions and graphs

24. International Studies Charter School.
Mrs. Rivas
International Studies Charter School.
Section 2-7
Absolute value Functions and graphs
Step # 1 Make a table.
Step # 2 Graph.
Step # 3 Use the location of the vertex to see how to translate the parent function.
𝒇 𝒙 = 𝒙 −𝟒
Since the vertex is at 𝟎,−𝟒 , you translated the graph of 𝒚= 𝒙 down 4 units.

25. International Studies Charter School.
Mrs. Rivas
International Studies Charter School.
Section 2-7
Absolute value Functions and graphs x
𝒚= 𝒙 +𝟐 −𝟑 𝟓 −𝟐 𝟒 −𝟏 𝟑 𝟎 𝟐 𝟏
Since the vertex is at 𝟎,𝟐 , you translated the graph of 𝒚= 𝒙 +𝟐 up 2 units.

26. International Studies Charter School.
Mrs. Rivas
International Studies Charter School.
Section 2-7
Absolute value Functions and graphs

27. International Studies Charter School.
Mrs. Rivas
International Studies Charter School.
Section 2-7
Absolute value Functions and graphs
+ left
+ up

28. International Studies Charter School.
Mrs. Rivas
International Studies Charter School.
Section 2-7
Absolute value Functions and graphs
- right
+ up

29. International Studies Charter School.
Mrs. Rivas
International Studies Charter School.
Section 2-7
Absolute value Functions and graphs
𝒚=𝒂 𝒙−𝒉 +𝒌
𝒗𝒆𝒓𝒕𝒆𝒙

30. International Studies Charter School.
Mrs. Rivas
International Studies Charter School.
Section 2-7
Absolute value Functions and graphs
Compare:
𝒚=𝟑 𝒙−𝟐 +𝟒
𝒂=𝟑
𝒉=𝟐
𝒌=𝟒
𝒚=𝒂 𝒙−𝒉 +𝒌
𝑽𝒆𝒓𝒕𝒆𝒙 = 𝟐,𝟒 𝒂𝒏𝒅 𝒕𝒉𝒆 𝒂𝒙𝒊𝒔 𝒐𝒇 𝒔𝒚𝒎𝒎𝒆𝒕𝒓𝒚𝒊𝒔 𝒙=𝟐
The parent function 𝒚=|𝒙| is translated right 2 unit, and translated up 4 units.

31. International Studies Charter School.
Mrs. Rivas
International Studies Charter School.
Section 2-7
Absolute value Functions and graphs
𝑽𝒆𝒓𝒕𝒆𝒙 = 𝟏,−𝟑 𝒂𝒏𝒅 𝒕𝒉𝒆 𝒂𝒙𝒊𝒔 𝒐𝒇 𝒔𝒚𝒎𝒎𝒆𝒕𝒓𝒚𝒊𝒔 𝒙=𝟏
The parent function 𝒚=|𝒙| is translated right 1 unit, then down 3 units.

32. International Studies Charter School.
Mrs. Rivas
International Studies Charter School.
Section 2-7
Absolute value Functions and graphs
Step #1: Identify the vertex.
𝑇ℎ𝑒 𝒗𝒆𝒓𝒕𝒆𝒙 𝑖𝑠 −𝟏, 𝟒 , 𝒉=−𝟏 𝑎𝑛𝑑 𝒌=𝟒
Step #2: Identify the slope (a).
𝑇ℎ𝑒 𝒔𝒍𝒐𝒑𝒆 𝑖𝑠 − 𝟏 𝟑 , so a = − 𝟏 𝟑
Step #3: Write the equation.
𝒚=𝒂 𝒙−𝒉 +𝒌
𝒚=− 𝟏 𝟑 𝒙−(−𝟏) +𝟒
𝒚=− 𝟏 𝟑 𝒙+𝟏 +𝟒

33. International Studies Charter School.
Mrs. Rivas
International Studies Charter School.
Section 2-7
Absolute value Functions and graphs
𝒚= 𝟏 𝟒 𝒙−𝟐 −𝟏

34. International Studies Charter School.
Mrs. Rivas
International Studies Charter School.
Section 2-7
Absolute value Functions and graphs

35. Mrs. Rivas 𝑽𝒆𝒓𝒕𝒆𝒙 (−𝟒, −𝟑) 𝒂𝒙𝒊𝒔 𝒐𝒇 𝒔𝒚𝒎𝒎𝒆𝒕𝒓𝒚 𝒙= −𝟒 𝑽𝒆𝒓𝒕𝒆𝒙 (𝟑, 𝟗)
Ida S. Baker H.S.
𝑽𝒆𝒓𝒕𝒆𝒙 (−𝟒, −𝟑)
𝒂𝒙𝒊𝒔 𝒐𝒇 𝒔𝒚𝒎𝒎𝒆𝒕𝒓𝒚 𝒙= −𝟒
𝑽𝒆𝒓𝒕𝒆𝒙 (𝟑, 𝟗)
𝒂𝒙𝒊𝒔 𝒐𝒇 𝒔𝒚𝒎𝒎𝒆𝒕𝒓𝒚 𝒙=𝟑

36. Mrs. Rivas 𝒚=𝒂 𝒙−𝒉 +𝒌 a: 𝒉, 𝒌 = 𝟔, 𝟎 𝒚=𝒂 𝒙−𝒉 +𝒌 a: 𝒉, 𝒌 = −𝟔, 𝟎
Ida S. Baker H.S.
Example:
Graph the absolute value equation
1. 𝒚= 𝟑 𝟐 𝒙−𝟔
𝒚=𝒂 𝒙−𝒉 +𝒌
𝟑 𝟐 a:
𝒉, 𝒌 =
𝟔, 𝟎
2. 𝒚=− 𝟑 𝟐 𝒙+𝟔
𝒚=𝒂 𝒙−𝒉 +𝒌
− 𝟑 𝟐 a:
𝒉, 𝒌 =
−𝟔, 𝟎

39. International Studies Charter School.
Mrs. Rivas
International Studies Charter School.
Practice 2-6 and 2-7