1. 4.3 Graphing Quadratic Equations and its Transformations
Ms. Lilian Albarico

2. Quadratic Function
The graph produced by y = x2 is a curve called a parabola. The parabola is a symmetrical curve that has one axis of symmetry. The axis of symmetry is a vertical line that passes through the turning point of the parabola. The turning point is called the vertex. The equation of the axis of symmetry is x = the value of the x co-ordinate of the vertex. Every parabola that is graphed is this standard curve with transformations performed on it.

3. Quadratic Function x y

4. Transformations: Vertical Reflection (Reflection in the x-axis)
Vertical Stretch
Vertical Translation/Shift
Horizontal Translation/Shift

5. Key Terms:
Mapping notation – a notation that describes how a graph and its image are related
x > (x, y)
reflection in x-axis – a transformation that vertically flips the graph of a curve in the x-axis
vertical stretch – a transformation that describes how the y-values of a curve are stretched by a scale factor 
vertical translation – a transformation that describes the number of units and the direction that a curve moves vertically from the origin
horizontal translation – a transformation that describes the number of units and the direction that a curve moves horizontally from the origin

6. Vertical Reflection
Vertical Reflection (V.R.) – a transformation in which a plane figure flips over vertically and has a horizontal axis of reflection. Also called as Reflection in x-axis.

7. Vert.ical Reflection

8. Vertical Reflection
y=x2  (x, y)
y=-x2  (x, -y)

9. Vertical Reflection
Draw the following quadratic equations, then write the mapping notations and include a table with 5 values:
f(x) = x and f(x) = x2
f(x) = 4x and f(x) = 4x2
f(x) = 1/4x2 and f(x) = 1/4x2

10. Vertical Stretch
Vertical Stretch (V.S.) – a transformation that describes how the y – value is stretched by a scale factor (the number times y)

11. Vertical Stretch

12. Vertical Stretch
1/2x2  (x, 2y)
2x2  (x, 1/2y)

13. Vertical Stretch
Draw the following quadratic equations, then write the mapping notation and include a table with 5 data values:
f(x) = x y = x /3y= x2
3y = x y= x /4y = x /4y = x2

14. Homework
CYU # 5 on page 175
CYU # 8, 9, 10,11 ,12 , and 13 on pages

15. Vertical Translation
Vertical Translation (V.T.) - a transformation that describes the number of units and direction that a curve has moved vertically.

16. Vertical Translation

17. Vertical Translation
x2 + 1 (x, y-1)
x2-1  (x, y+1)

18. Vertical Translation
Draw the following quadratic equations, then write the mapping notation and include a table with 5 data values:
y = x y + 2 = x y-2 = x2
y-3 = x2 (y+3)= -x2 y-1/2 = x2 y+1/2 = 2x2

19. Horizontal Translation
Horizontal Translation (H.T.) – a transformation that describes the number of units and direction that a curve has moved horizontally.

20. Horizontal Translation

21. Horizontal Translation
(x+1)2  (x-1, y)
(x-1)2  (x+1, y)

22. Horizontal Translation
Draw the following quadratic equations, then write the mapping notation and include a table with 5 data values:
y = x y = (x+3) y = (x-4)2
y = 2(x+3)2 -y= (x-2) y = (x-2)2 y-1 = (x+1)2

23. HOMEWORK CYU #14 AND 14 ON PAGE 179

24. SUMMARY

25. Learning Check:
When you see a –y in the equation then the image is a _________________ in the x-axis.
“When I see a 3y in the equation, then I know that there is a vertical stretch of ___________.”
The vertical stretch factor is the _________ of the coefficient of y.
“When I see y+2 in the equation, then I know that there is vertical translation of ____ units _____”.
“When I see y-3 in the equation, then I know that there is vertical translation of ____ units _____”.
The vertical translation is the ___________of the number being added to the y.
“When I see x+3 in the equation, then I know that there is horizontal translation by ____ units to the _____”.
“When I see x-4 in the equation, then I know that there is horizontal translation by ____ units to the _____”.
The horizontal translation is the ___________ of the number being added to the x.”

26. Let’s combine all transformations!

27. -y = (x+2)2 -2y = (x-1)2 y = (x-3)2
1) Graph the following quadratic functions and write down their mapping notations:
-y = (x+2)2
-2y = (x-1)2
y = (x-3)2

28. (x , y)  (x+2, y) (x, y)  (x+6, y-1) (x, y)  (x-1/2, y – 3/2)
2) Based from the mapping notation below, write the quadratic equation and the draw its graph transformation:
(x , y)  (x+2, y)
(x, y)  (x+6, y-1)
(x, y)  (x-1/2, y – 3/2)
(x, y)  (x-2, -2y)

29. Absolute Value Functions

30. Absolute Value Functions
absolute value – the absolute value of a number is the number without its sign
|n| = n, |-n| = n.
If absolute value is related to a number line, then it indicates the measure of the distance between a point and its origin without reference to direction. Absolute value functions are graphed using y = |x| after the transformations have been applied.

31. y – q = |x|  (x, y+q) For example: y-3 = |x| y+5= |x|
VERTICAL TRANSLATION
y – q = |x|  (x, y+q) For example: y-3 = |x| y+5= |x|

33. HORIZONTAL TRANSLATION
y = |x-p|  (x+p, y) For example: y = |x-3| y= |x+4|

35. cy = |x|  (x, 1/3y) For example: 1/2y = |x| 2y= |x|
VERTICAL STRETCH
cy = |x|  (x, 1/3y) For example: 1/2y = |x| 2y= |x|

37. -y = |x|  (x, -y) For example: y = -1/10|x| 1/10y= |x|
REFLECTION in X-AXIS
-y = |x|  (x, -y) For example: y = -1/10|x| 1/10y= |x|

39. Putting it All Together

40. -1/2 (y+3) = |x+5| y-3 = |x+3| 1/4y= |x-2|
1) Graph the following absolute value functions and write its mapping notation:
-1/2 (y+3) = |x+5|
y-3 = |x+3|
1/4y= |x-2|

41. (x , y)  (x+1, y) (x, y)  (x, y-4) (x, y)  (x-3, y + 3)
2) Write the absolute value functions of the following mapping notation and graph its transformation:
(x , y)  (x+1, y)
(x, y)  (x, y-4)
(x, y)  (x-3, y + 3)
(x, y)  (-x, y+5)

42. Assignment
You will draw a shape using different kinds of graphs – linear, quadratic, exponential, and absolute value functions. There should only be one point of origin. Make sure your graphs are clear and accurate. Be as creative as you can.
Below your graph:
A) State the domain and range of your different graphs.
B) If it is either quadratic or absolute value functions, write its mapping notations.
Deadline : December 26, 2012

43. Homework FOCUS QUESTIONS # 30 and 31 on page 186.
CYU # 33, 34, 35, 36 and 37 on pages