1. 10.1: Quadratic Equations and Functions
Do NOW
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10.1: Quadratic Equations and Functions

2. 10.1: Quadratic Equations and Functions
SECTION 10.2
Breaking Down a
Quadratic
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10.1: Quadratic Equations and Functions

3. 10.1: Quadratic Equations and Functions
Graph using a Table
Pick five points and plug into the equation
Connect the dots to make the shape
Make sure to accurately label the zeros and the y-intercept, if applicable
Make sure it is the correct shape
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10.1: Quadratic Equations and Functions

4. 10.1: Quadratic Equations and Functions
The Change in C
For shifts of c of ax2 + bx + c = y, the change of the c, the y–intercept shifts up or down
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10.1: Quadratic Equations and Functions

5. 10.1: Quadratic Equations and Functions
Example 1
Graph y = x2 x y –2 –1 1 2
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10.1: Quadratic Equations and Functions

6. 10.1: Quadratic Equations and Functions
Example 1
Graph y = x2 x y –2 –1 1 2
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10.1: Quadratic Equations and Functions

7. 10.1: Quadratic Equations and Functions
Example 2
Graph y = x2 + 2 and describe the transformation from the parent function. x y –2 –1 1 2
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10.1: Quadratic Equations and Functions

8. 10.1: Quadratic Equations and Functions
The Change in C
For shifts of c of ax2 + bx + c = y, the change of the c, the y–intercept shifts up or down
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10.1: Quadratic Equations and Functions

9. 10.1: Quadratic Equations and Functions
Your Turn
Graph y = x2 – 3 and describe the transformation from the parent function. x y –2 –1 1 2
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10.1: Quadratic Equations and Functions

10. 10.1: Quadratic Equations and Functions
Example 3
Graph y = 2x2 and describe the transformation from the parent function. x y –2 –1 1 2
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10.1: Quadratic Equations and Functions

11. 10.1: Quadratic Equations and Functions
Example 3
Graph y = 2x2 and describe the transformation from the parent function. x y –2 –1 1 2
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10.1: Quadratic Equations and Functions

12. 10.1: Quadratic Equations and Functions
The Change in A
For shifts of a of ax2 + bx + c = y, the bigger the a, the smaller the graph gets.
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10.1: Quadratic Equations and Functions

13. 10.1: Quadratic Equations and Functions
Example 4
Graph y = (1/2)x2 x y –2 –1 1 2
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10.1: Quadratic Equations and Functions

14. 10.1: Quadratic Equations and Functions
Example 5
Graph y = –(1/2)x2 x y –2 –1 1 2
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10.1: Quadratic Equations and Functions

15. 10.1: Quadratic Equations and Functions
Your Turn
Graph y = –x2 + 4 x y –2 –1 1 2
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10.1: Quadratic Equations and Functions

16. 10.1: Quadratic Equations and Functions
Intro to Quadratics
Quadratic Function is viewed as, ax2 + bx + c = 0, where a ≠ 0
Parabola is an u–shaped graph
If a is positive, it opens up
If a is negative, it opens down
Vertex is the highest or lowest point of the graph
Axis of Symmetry is the vertical line passing though the vertex (x of vertex) represented by a DOTTED–line
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10.1: Quadratic Equations and Functions

17. 10.1: Quadratic Equations and Functions
Definitions
Roots: The solutions to the equation Also known as solutions
X-intercepts: Point(s) where the graph crosses the x–axis. Also known as roots and x-intercepts.
Ordered Pair: (2, 0)
Vertex: Minimum or maximum value
Axis of Symmetry: Line that separates the graph in half; always is x equals
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10.1: Quadratic Equations and Functions

18. 10.1: Quadratic Equations and Functions
Definitions
Determine:
How it opens: Does it open up or down?
Roots: Where does it cross the x-axis?
Y-Intercept: Where does it cross the y-axis?
Vertex: The highest/lowest part of the graph
Axis of Symmetry: Equation of which the X of the vertex
Domain: The set of all x-values.
Range: The set of all y-values.
Minimum/Maximum: The highest or lowest part of the graph
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10.1: Quadratic Equations and Functions

19. 10.1: Quadratic Equations and Functions
Worksheet Key –5
–5 < x < 4 –2
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10.1: Quadratic Equations and Functions

20. 10.1: Quadratic Equations and Functions
Worksheet Key
7, 11.5}
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10.1: Quadratic Equations and Functions

21. 10.1: Quadratic Equations and Functions
Definitions
Quadratic Equation is viewed as, ax2 + bx + c = 0, where a ≠ 0
Parent Functions of Parabola:
Change of a Change of c
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10.1: Quadratic Equations and Functions

22. 10.1: Quadratic Equations and Functions
Example 5
Graph y = x2 – 9 and identify:
Graph the function and label the vertex, roots, axis of symmetry, and y-intercept
How it opens: Does it open up or down?
Roots: Where does it cross the x-axis?
Y-Intercept: Where does it cross the y-axis?
Vertex: The highest/lowest part of the graph
Line of Symmetry: Equation of which the X of the vertex
Domain: The set of all x-values.
Range: The set of all y-values.
Minimum/Maximum: The highest or lowest part of the graph
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10.1: Quadratic Equations and Functions

23. 10.1: Quadratic Equations and Functions
Example 5
Graph y = x2 – 9 and identify: x y –3 –2 –1 1 2 3
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10.1: Quadratic Equations and Functions

24. 10.1: Quadratic Equations and Functions
Example 5
Graph y = x2 – 9 and identify:
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10.1: Quadratic Equations and Functions

25. 10.1: Quadratic Equations and Functions
Example 5
Graph y = x2 – 9 and identify:
How it opens
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10.1: Quadratic Equations and Functions

26. 10.1: Quadratic Equations and Functions
Example 5
Graph y = x2 – 9 and identify:
Roots:
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10.1: Quadratic Equations and Functions

27. 10.1: Quadratic Equations and Functions
Example 5
Graph y = x2 – 9 and identify:
Y-intercept
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10.1: Quadratic Equations and Functions

28. Example 5 Graph y = x2 – 9 and identify:
Y-intercept (with a calculator)
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10.1: Quadratic Equations and Functions

29. 10.1: Quadratic Equations and Functions
Example 5
Graph y = x2 – 9 and identify:
The Vertex
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10.1: Quadratic Equations and Functions

30. 10.1: Quadratic Equations and Functions
Example 5
Graph y = x2 – 9 and identify:
The Vertex
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10.1: Quadratic Equations and Functions

31. Example 5 Graph y = x2 – 9 and identify: Axis of Symmetry
Line of Symmetry is the X of the vertex.
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10.1: Quadratic Equations and Functions

32. Domain (as an inequality) 10.1: Quadratic Equations and Functions
Example 5
Graph y = x2 – 9 and identify:
Domain (as an inequality)
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10.1: Quadratic Equations and Functions

33. Domain (with a calculator) 10.1: Quadratic Equations and Functions
Example 5
Graph y = x2 – 9 and identify:
Domain (with a calculator)
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10.1: Quadratic Equations and Functions

34. Range (as an inequality) 10.1: Quadratic Equations and Functions
Example 5
Graph y = x2 – 9 and identify:
Range (as an inequality)
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10.1: Quadratic Equations and Functions

35. 10.1: Quadratic Equations and Functions
Example 5
Graph y = x2 – 9 and identify:
Minimum/Maximum
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10.1: Quadratic Equations and Functions

36. 10.1: Quadratic Equations and Functions
Example 5
Graph y = x2 – 9 and identify:
Axis of Symmetry x y –3 –2 –1 1 2 3
Roots
Roots
Y-intercept
Vertex
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10.1: Quadratic Equations and Functions

37. 10.1: Quadratic Equations and Functions
Example 6
Graph y = x2 – 4 and identify:
Graph
How it opens
Roots
Y-Intercept
Vertex
Line of Symmetry
Domain
Range
Minimum/Maximum
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10.1: Quadratic Equations and Functions

38. 10.1: Quadratic Equations and Functions
Your Turn
Graph y = x2 – 1 and identify:
Graph
How it opens
Roots
Y-Intercept
Vertex
Line of Symmetry
Domain
Range
Minimum/Maximum
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10.1: Quadratic Equations and Functions

39. 10.1: Quadratic Equations and Functions
Example 7
Graph y = 2x2 – 8 and identify:
Graph the function and label the vertex, roots, axis of symmetry, and y-intercept
How it opens: Does it open up or down?
Roots: Where does it cross the x-axis?
Y-Intercept: Where does it cross the y-axis?
Vertex: The highest/lowest part of the graph
Line of Symmetry: Equation of which the X of the vertex
Domain: The set of all x-values.
Range: The set of all y-values.
Minimum/Maximum: The highest or lowest part of the graph
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10.1: Quadratic Equations and Functions

40. 10.1: Quadratic Equations and Functions
Example 7
Graph y = 2x2 – 8 and identify: x y –3 –2 –1 1 2 3
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10.1: Quadratic Equations and Functions

41. 10.1: Quadratic Equations and Functions
Example 7
Graph y = 2x2 – 8 and identify:
Graph
How it opens
Roots
Y-Intercept
Vertex
Line of Symmetry
Domain
Range
Minimum/Maximum
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10.1: Quadratic Equations and Functions

42. 10.1: Quadratic Equations and Functions
Example 8
Graph y = (–1/2)x2 + 4 and identify:
Graph the function and label the vertex, roots, axis of symmetry, and y-intercept
How it opens: Does it open up or down?
Y-Intercept: Where does it cross the y-axis?
Vertex: The highest/lowest part of the graph
Line of Symmetry: Equation of which the X of the vertex
Domain: The set of all x-values.
Range: The set of all y-values.
Minimum/Maximum: The highest or lowest part of the graph
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10.1: Quadratic Equations and Functions

43. 10.1: Quadratic Equations and Functions
Example 8
Graph y = (–1/2)x2 + 4 and identify: x y –3 –2 –1 1 2 3
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10.1: Quadratic Equations and Functions

44. 10.1: Quadratic Equations and Functions
Example 8
Graph y = (–1/2)x2 + 4 and identify:
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10.1: Quadratic Equations and Functions

45. 10.1: Quadratic Equations and Functions
Example 8
Graph y = (–1/2)x2 + 4 and identify:
Graph
How it opens
Y-Intercept
Vertex
Line of Symmetry
Domain
Range
Minimum/Maximum
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10.1: Quadratic Equations and Functions

46. 10.1: Quadratic Equations and Functions
Your Turn
Graph y = (1/2)x2 – 4 and identify:
Graph the function and label the vertex, roots, axis of symmetry, and y-intercept {Domain: –4, –2, 0, 2, 4}
How it opens: Does it open up or down?
Y-Intercept: Where does it cross the y-axis?
Vertex: The highest/lowest part of the graph
Line of Symmetry: Equation of which the X of the vertex
Domain: The set of all x-values.
Range: The set of all y-values.
Minimum/Maximum: The highest or lowest part of the graph
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10.1: Quadratic Equations and Functions

47. 10.1: Quadratic Equations and Functions
Your Turn
Graph y = (1/2)x2 – 4 and identify: x y –4 –2 2 4
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10.1: Quadratic Equations and Functions

48. 10.1: Quadratic Equations and Functions
Your Turn
Graph y = (1/2)x2 – 4 and identify:
Graph
How it opens
Roots
Y-Intercept
Vertex
Line of Symmetry
Domain
Range
Minimum/Maximum
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10.1: Quadratic Equations and Functions

49. 10.1: Quadratic Equations and Functions
Writing Equations
Use the information given to create the equation from the original equation
If it is shift on the y-intercept, it effects the c
If it is shift on the size, it effects the a
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10.1: Quadratic Equations and Functions

50. 10.1: Quadratic Equations and Functions
Example 9
Given the function y = x2 + 2, describe the shift in the vertex of the parabola if, in the function, 2 is changed to –2?
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10.1: Quadratic Equations and Functions

51. 10.1: Quadratic Equations and Functions
Example 10
What is the equation of the function that would be shifted 11 units up from the graph of f(x) = x2 – 8
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10.1: Quadratic Equations and Functions

52. 10.1: Quadratic Equations and Functions
Example 11
What is the equation of the function that would be shifted 11 units up from the graph of f(x) = x2 – 8?
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10.1: Quadratic Equations and Functions

53. 10.1: Quadratic Equations and Functions
Your Turn
What is the equation of the function that would be shifted 11 units down from the graph of f(x) = x2 + 3?
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10.1: Quadratic Equations and Functions

54. 10.1: Quadratic Equations and Functions
Assignment
Worksheet
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10.1: Quadratic Equations and Functions