1. Quadratic Functions and Transformations Lesson 4-1
Algebra 2
Quadratic Functions and Transformations
Lesson 4-1

2. Goals Goal Rubric To identify and graph quadratic functions.
Level 1 – Know the goals.
Level 2 – Fully understand the goals.
Level 3 – Use the goals to solve simple problems.
Level 4 – Use the goals to solve more advanced problems.
Level 5 – Adapts and applies the goals to different and more complex problems.

3. Vocabulary Parabola Quadratic Function Vertex Form Axis of Symmetry
Vertex of the Parabola
Minimum Value
Maximum Value

4. Big Idea: Function and Equivalence
Essential Question
Big Idea: Function and Equivalence
What is the vertex form of a quadratic function?

5. Quadratic Function
In Chapters 2 and 3, you studied linear functions of the form f(x) = mx + b.
A quadratic function in two variables can be written in standard form f(x) = ax2 + bx + c, where a, b, and c are real numbers and a ≠ 0.
The simplest quadratic function f(x) = x2 is the parent function.
In a quadratic function, the variable is always squared.

6. Parabola
The graph of a quadratic function is a curve called a parabola.
A parabola is a U-shaped curve as shown at the right.
To graph a quadratic function, generate enough ordered pairs to see the shape of the parabola. Then connect the points with a smooth curve.

7. Vertex The highest or lowest point on a parabola is the vertex.
If a parabola opens upward, the vertex is the lowest point.
If a parabola opens downward, the vertex is the highest point.
Vertex is the highest point
Vertex is the lowest point

8. Axis of Symmetry
The vertical line that divides a parabola into two symmetrical halves is the axis of symmetry.
The axis of symmetry always passes through the vertex of the parabola.
Axis of symmetry
Vertex
Axis of symmetry
Vertex

9. Summary

10. Quadratic Function
A second form of the quadratic function is vertex form.
The vertex form of a quadratic function is f(x) = a (x – h)2 + k where a ≠ 0.
It is called vertex form because it contains the vertex (h, k).

11. Quadratic Function Transformations
Transformations in the horizontal direction are done to the input of a function (x).
Transformations in the vertical direction are done to the output of a function (f(x)).
Parent Quadratic Function f(x) = x2.
Output
f(x) = x2
Input

12. Translations

13. Horizontal Translation of the Form f(x) = (x + h)2
Example: Graph the functions on one coordinate plane. x y 2 4 6 8 2 4 6 8
f(x) = x 2
g(x) = (x + 3)2
g(x) = (x + 3)2
f(x) = x 2 x
f(x)
 2 4 1 1 2 x
g(x)
 2 1 1 4 9 16 2 25
Notice that the graph of g(x) is the graph of f(x) shifted 3 units to the left.

14. Horizontal Translation of the Form f(x) = (x – h)2
Example: Graph the functions on one coordinate plane.
f(x) = x 2 x y 2 4 6 8 2 4 6 8
g(x) = (x – 1)2
g(x) = (x – 1)2
f(x) = x 2 x
f(x)
 2 4 1 1 2 x
g(x)
 2 9 1 4 1 2
Notice that the graph of g(x) is the graph of f(x) shifted 1 unit to the right.

15. Vertical Translation of the Form f(x) = x 2 + ky 2 4 6 8 2 4 6 8
Example: Graph the functions on one coordinate plane.
g(x) = x 2 + 3
f(x) = x 2
g(x) = x 2 + 3
f(x) = x 2 x
f(x)
 2 4 1 1 2 x
g(x)
 2 7 1 4 3 1 2
Notice that the graph of g(x) is the graph of f(x) shifted 3 units upward.

16. Vertical Translation of the Form f(x) = x 2 – k
Example: Graph the functions on one coordinate plane. x y 2 4 6 8 2 4 6 8
f(x) = x 2
g(x) = x 2 – 2
f(x) = x 2
g(x) = x 2 – 2 x
f(x)
 2 4 1 1 2 x
g(x)
 2 2 1
 1 1
Notice that the graph of g(x) is the graph of f(x) shifted 2 units downward.

17. Combining Translations
Example: Graph the functions on one coordinate plane. x y 2 4 6 8 2 4 6 8
g(x) = (x + 1)2 + 2
f(x) = x 2
g(x) = (x + 1)2 + 2
f(x) = x 2 x
f(x)
 2 4 1 1 2 x
g(x)
 2 3 1 2 1 6 11
Notice that the graph of g(x) is the graph of f(x) shifted 1 unit to the left and 2 units upward.

18. Example:
Use the graph of f(x) = x2 as a guide, describe the transformations and then graph each function.
g(x) = (x – 2)2 + 4
Identify h and k.
g(x) = (x – 2)2 + 4 h k
Because h = 2, the graph is translated 2 units right. Because k = 4, the graph is translated 4 units up. Therefore, g is f translated 2 units right and 4 units up.

19. Example:
Use the graph of f(x) = x2 as a guide, describe the transformations and then graph each function.
g(x) = (x + 2)2 – 3
Identify h and k.
g(x) = (x – (–2))2 + (–3) h k
Because h = –2, the graph is translated 2 units left. Because k = –3, the graph is translated 3 units down. Therefore, g is f translated 2 units left and 4 units down.

20. Your Turn:
Using the graph of f(x) = x2 as a guide, describe the transformations and then graph each function.
g(x) = x2 – 5
Identify h and k.
g(x) = x2 – 5 k
Because h = 0, the graph is not translated horizontally. Because k = –5, the graph is translated 5 units down. Therefore, g is f is translated 5 units down.

21. Your Turn:
Use the graph of f(x) =x2 as a guide, describe the transformations and then graph each function.
g(x) = (x + 3)2 – 2
Identify h and k.
g(x) = (x – (–3)) 2 + (–2) h k
Because h = –3, the graph is translated 3 units left. Because k = –2, the graph is translated 2 units down. Therefore, g is f translated 3 units left and 2 units down.

22. More Transformations
Recall that functions can also be reflected, stretched, or compressed.
This includes quadratic functions.

23. Quadratic Function Transformations

24. Vertical Stretch of the Form f(x) = ax 2
Example: Graph the functions on one coordinate plane. x y 2 4 6 8 2 4 6 8
g(x) = 2x 2
f(x) = x 2
g(x) = 2x 2
f(x) = x 2 x
f(x)
 2 4 1 1 2 x
g(x)
 2 8 1 2 1
Notice that the graph of g(x) is narrower than the graph of f(x).

25. Vertical Compression of the Form f(x) = ax 2
Example: Graph the functions on one coordinate plane.
f(x) = x 2 x y 2 4 6 8 2 4 6 8
g(x) = x 2
f(x) = x 2
g(x) = x 2 x
f(x)
 2 4 1 1 2 1 2 1
 2
g(x) x
Notice that the graph of g(x) is wider than the graph of f(x).

26. Reflection Over the x-axis of the Form f(x) = -x 2
Example: Graph the functions on one coordinate plane. x y 2 4 6 8 2 4 6 8
g(x) = –x 2 x
g(x)
 2
 4 1
 1 1 2
f(x) = x 2
f(x) = x 2 x
f(x)
 2 4 1 1 2
g(x) = – x 2
Notice that the graph of g(x) is a reflection of the graph of f(x) over the x-axis.

27. Example:
Using the graph of f(x) = x2 as a guide, describe the transformations and then graph each function. 1 ( )
g x =- x 2 4
Because a is negative, g is a reflection of f across the x-axis.
Because |a| = , g is a vertical compression of f by a factor of .

28. Example:
Using the graph of f(x) = x2 as a guide, describe the transformations and then graph each function.
g(x) = 3x2
Because a = 3 , g is a vertical stretch of f by a factor of 3.

29. Your Turn:
Using the graph of f(x) = x2 as a guide, describe the transformations and then graph each function.
g(x) = 2x2
Because a = 2, g is a vertical stretch of f by a factor of 2.

30. Your Turn:
Using the graph of f(x) = x2 as a guide, describe the transformations and then graph each function.
g(x) = – x2
Because a is negative, g is a reflection of f across the x-axis.
Because |a| = , g is a vertical compression of f by a factor of .

31. Minimums and Maximums
If a parabola opens upward, it has a lowest point, a minimum. If a parabola opens downward, it has a highest point, a maximum. This minimum or maximum is the vertex of the parabola.
If a > 0, the parabola opens upward. The vertex is a minimum point.
If a < 0, the parabola opens upward. The vertex is a maximum point.
Vertex is the maximum
Vertex is the minimum
Opens up
a > 0
Opens down
a < 0

32. Minimums and Maximums

33. More Vertex Form
The parent function f(x) = x2 has its vertex at the origin.
You can identify the vertex of other quadratic functions by analyzing the function in vertex form.
The vertex form of a quadratic function is f(x) = a(x – h)2 + k, where a, h, and k are constants and (h, k) is the vertex.

34. Because the vertex is translated h horizontal units and k vertical from the origin, the vertex of the parabola is at (h, k).
When the quadratic parent function f(x) = x2 is written in vertex form, y = a(x – h)2 + k, a = 1, h = 0, and k = 0.
Helpful Hint

35. Summary of Transformations
f(x) = ± a(x – h)2 + k
Vertical Translation
Same direction of sign
+ up
- down
Reflection about
the x-axis.
+ opens upward
- opens downward
Vertical stretch/compress
a>1 narrower
0<a<1 wider
Horizontal Translation
Opposite direction of sign
+ to the left
- to the right

37. Example:
Use the description to write the quadratic function in vertex form.
The parent function f(x) = x2 is vertically stretched by a factor of and then translated 2 units left and 5 units down to create g.
Step 1 Identify how each transformation affects the constant in vertex form.
Vertical stretch by : 4 3 a =
Translation 2 units left: h = –2
Translation 5 units down: k = –5

38. Example: continued Step 2 Write the transformed function.
g(x) = a(x – h)2 + k
Vertex form of a quadratic function
= (x – (–2))2 + (–5)
Substitute for a, –2 for h, and –5 for k.
= (x + 2)2 – 5
Simplify.
g(x) = (x + 2)2 – 5

39. Your Turn:
Use the description to write the quadratic function in vertex form.
The parent function f(x) = x2 is vertically compressed by a factor of and then translated units right and 4 units down to create g.
Step 1 Identify how each transformation affects the constant in vertex form.
Vertical compression by : a =
Translation 2 units right: h = 2
Translation 4 units down: k = –4

40. Continued Step 2 Write the transformed function. g(x) = a(x – h)2 + k
Vertex form of a quadratic function
= (x – 2)2 + (–4)
Substitute for a, 2 for h, and –4 for k.
= (x – 2)2 – 4
Simplify.
g(x) = (x – 2)2 – 4

41. Your Turn:
Use the description to write the quadratic function in vertex form.
The parent function f(x) = x2 is reflected across the x-axis and translated 5 units left and 1 unit up to create g.
Step 1 Identify how each transformation affects the constant in vertex form.
Reflected across the x-axis: a is negative
Translation 5 units left: h = –5
Translation 1 unit up: k = 1

42. Continued Step 2 Write the transformed function. g(x) = a(x – h)2 + k
Vertex form of a quadratic function
= –(x –(–5))2 + (1)
Substitute –1 for a, –5 for h, and 1 for k.
= –(x +5)2 + 1
Simplify.
g(x) = –(x +5)2 + 1

43. Example:
What is an equation of the function shown in the graph?

44. Solution:
The vertex of the parabola is at (2, –3), so h = 2 and k = –3. Since the graph passes through (0, –1), let x = 0 and y = –1. Substitute these values into the vertex form of the equation and solve for a.

45. Solution:
Vertex form
Substitute –1 for y, 0 for x, 2 for h, and –3 for k.
Simplify.
Add 3 to each side.
Divide each side by 4.

46. Solution:
The equation of the parabola in vertex form is

47. Your Turn:
What is an equation of the function shown in the graph?
𝑓 𝑥 =−3 𝑥 A B C D

48. Example: Appliication
On Earth, the distance d in meters that a dropped object falls in t seconds is approximated by d(t)= 4.9t2. On the moon, the corresponding function is dm(t)= 0.8t2. What kind of transformation describes this change from d(t)= 4.9t2, and what does the transformation mean?
Examine both functions in vertex form.
d(t)= 4.9(t – 0)2 + 0
dm(t)= 0.8(t – 0)2 + 0

49. Example: continued
The value of a has decreased from 4.9 to 0.8. The decrease indicates a vertical compression.
Find the compression factor by comparing the new a-value to the old a-value.
a from d(t)
a from dm(t) =
0.8
4.9 
0.16
The function dm represents a vertical compression of d by a factor of approximately Because the value of each function approximates the time it takes an object to fall, an object dropped from the moon falls about 0.16 times as fast as an object dropped on Earth.

50. Your Turn:
The minimum braking distance d in feet for a vehicle on dry concrete is approximated by the function (v) = 0.045v2, where v is the vehicle’s speed in miles per hour.
The minimum braking distance dn in feet for a vehicle with new tires at optimal inflation is dn(v) = 0.039v2, where v is the vehicle’s speed in miles per hour. What kind of transformation describes this change from d(v) = 0.045v2, and what does this transformation mean?

51. Continued Examine both functions in vertex form.
d(v)= 0.045(t – 0)2 + 0
dn(t)= 0.039(t – 0)2 + 0
The value of a has decreased from to The decrease indicates a vertical compression.
Find the compression factor by comparing the new a-value to the old a-value. =
a from d(v)
a from dn(t)
0.039
0.045 13 15

52. Continued
The function dn represents a vertical compression of d by a factor of The braking distance will be less with optimally inflated new tires than with tires having more wear.
Check the graph. The graph of dn appears to be vertically compressed compared with the graph of d. 15 d dn

53. Graphing Using Transformations
f(x) = 2(x - 1)2 - 4
Stretch the graph of y = x2.

54. Graphing Using Transformations
f(x) = 2(x - 1)2 - 4
Vertically stretch the curve in #1 by a factor of 2, to sketch y = 2x2.

55. Graphing Using Transformations
f(x) = 2(x - 1)2 - 4
Move the curve in #2 horizontally one unit to the right to sketch y = 2(x – 1)2.

56. Graphing Using Transformations
f(x) = 2(x - 1)2 - 4
Move the curve in #3 vertically down 4 units to sketch y = 2(x – 1)2 -4.

57. Graphing Using Transformations
f(x) = 2(x - 1)2 - 4

58. Graphing a Quadratic Function in Vertex Form
Procedure: quadratic function in vertex form
Identify and plot the vertex (h, k). Find the axis of symmetry, x = h.
Find and plot two points on one side of the axis of symmetry.
Plot the corresponding points on the other side of the axis of symmetry.
Sketch the curve.

59. Example: Graph
Vertex Form

60. 1. Identify and plot the vertex (h, k)
1. Identify and plot the vertex (h, k). Find the axis of symmetry, x = h
Vertex

61. 2. Find and plot 2 points on one side of the axis of symmetry;
3. Plot the corresponding points on the other side of the axis of symmetry.
Axis of
Symmetry
x=1
4. Sketch the curve.

62. Example: Graph
Vertex Form

63. Vertex Axis of Symmetry x=1
1. Identify and plot the vertex (h, k). Find the axis of symmetry, x = h
Vertex
Axis of
Symmetry
x=1

64. 2. Find and plot 2 points on one side of the axis of symmetry;
3. Plot the corresponding points on the other side of the axis of symmetry.

65. 4. Sketch the curve.

66. Your Turn:
Graph y = –2(x – 1)2 + 3.
Solution:

67. Your Turn:
Graph y = 3(x + 2)
Solution:

68. Big Idea: Function and Equivalence
Essential Question
Big Idea: Function and Equivalence
What is the vertex form of a quadratic function?
The vertex form of a quadratic function is f(x) = a(x – h)2 + k, where a ). All quadratic functions are transformations of the parent function, f(x) = x2. Use vertex form to identify the transformations and graph a quadratic function.

69. Assignment
Section 4-1, Pg. 209 – 211; #1 – 6 all, 8 – 26 even, 30 – 54 even.