1. Identifying key Features on a parabola unit 3 day 1

2. KEY CONTENT:
F-IF.B.4: I can interpret key features of a graph. Key features include: intercepts, intervals where the function is increasing and decreasing, maximums and minimums, symmetries, end behavior.
F-IF.B.6: I can calculate the average rate of change of a function over a specific interval.
F-IF.A.1: I can determine the domain and
range of a graph of a function.

3. Quadratic Functions
Curved antennas, such as the ones shown to the left, are commonly used to focus microwaves and radio waves to transmit television and telephone signals, as well as satellite and spacecraft communication. The cross-section of the antenna is in the shape of a parabola, which can be described by a quadratic function.

4. Recognizing Characteristics of Parabolas
The graph of a quadratic function is a U-shaped curve called a parabola. One important feature of the graph is that it has an extreme point, called the vertex. If the parabola opens up, the vertex represents the lowest point on the graph, or the minimum value of the quadratic function. If the parabola opens down, the vertex represents the highest point on the graph, or the maximum value.
In either case, the vertex is a turning point on the graph. The graph is also symmetric with a vertical line drawn through the vertex, called the axis of symmetry.

5. Recognizing Characteristics of Parabolas
The y-intercept is the point at which the parabola crosses the y-axis. The 
x-intercepts are the points at which the parabola crosses the x-axis. If they exist, the x-intercepts represent the solutions, zeros, or roots, of the quadratic function. The zeros or roots can be found when y = 0. 

6. Recognizing Characteristics of Parabolas
No Real Solutions/Two Imaginary Solutions
Later in this unit you will learn how to algebraically find the solution(s) of a quadratic equation. Solutions are also called
x-intercepts, roots or zeros. When a quadratic function is graphically represented by a parabola, the direction the parabola opens up and the location of the vertex, in relation to the  x-axis, tell you the type and quantity of solutions the function contains.
One Real Solutions
Two Real Solutions

7. Mini Lesson #1 Identifying the domain and range of a quadratic function In interval notation
Interval notation is notation for representing an interval as a pair of numbers. The numbers are the endpoints of the interval. Parentheses and/or brackets are used to show whether the endpoints are excluded or included. 
Watch only specific portions of the videos below.
Video – Take Notes - (3:35-5:40)
Video – Take Notes - (7:30-8:30)

8. Mini Lesson #1 Identifying the domain and range of a quadratic function In interval notation
INDIVIDUAL PRACTICE
Using interval notation, identify the domain
and range of the quadratic function to the right.
Answer is revealed in the video (9:40-10:05)

9. Mini Lesson #2 Finding the end behavior of a quadratic function
The end behavior of a function is the behavior of the graph of f(x) as x approaches positive infinity or negative infinity. The degree and the leading coefficient of a polynomial function determine the end behavior of the graph.
Watch the entire video below. Take Notes.
2nd Degree & Positive Leading Coefficient
2nd Degree & Negative Leading Coefficient

10. Mini Lesson #2 Finding the end behavior of a quadratic function
INDIVIDUAL PRACTICE
Determine the end behavior of the quadratic
function to the right.

11. Mini Lesson #2 Finding the end behavior of a quadratic function
INDIVIDUAL PRACTICE
Determine the end behavior of the quadratic
function to the right.

12. Mini Lesson #3 identify the increasing and decreasing intervals
Video – Take Notes - (2:47-4:35)

13. Mini Lesson #3 identify the increasing and decreasing intervals
Quadratic functions increase and decrease
The function is increasing when x > 2.
The function is decreasing when x < 2.
The function does not increase or decrease when x = 2.
Increase:
Decrease:

14. Mini Lesson #3 identify the increasing and decreasing intervals
Quadratic functions increase and decrease
The function is increasing when x < 1.
The function is decreasing when x > 1.
The function does not increase or decrease when x = 1.
Increase:
Decrease:

15. Mini Lesson #3 identify the increasing and decreasing intervals
INDIVIDUAL PRACTICE
Determine the increasing and decreasing intervals
in the quadratic function to the right.

16. Mini Lesson #3 identify the increasing and decreasing intervals
INDIVIDUAL PRACTICE
Determine the increasing and decreasing intervals
in the quadratic function to the right.
Increase:
Decrease:

17. Mini Lesson #4 identify the intercepts of a quadratic function
Video – Take Notes - (0:00 - 2:35)

18. Mini Lesson #4 identify the intercepts of a quadratic function
INDIVIDUAL PRACTICE
Determine the intercepts of the function below.

20. Mini Lesson #4 identify the intercepts of a quadratic function
INDIVIDUAL PRACTICE
Determine the intercepts of the function below.

21. Guided practice Example 1
Identify the following characteristics of the parabola to the right.
vertex coordinates
equation to the axis of symmetry
y-intercept of the parabola
x-intercept of the parabola
domain in interval notation
range in interval notation
interval of increase
interval of decrease
average rat of change on the interval [0,3]
circle: maximum value or minimum value
end behavior

22. Guided practice Example 1
Identify the following characteristics of the parabola to the right.
vertex coordinates

23. Guided practice Example 1
Identify the following characteristics of the parabola to the right.
(2) equation to the axis of symmetry

24. Guided practice Example 1
Identify the following characteristics of the parabola to the right.
(3) y-intercept of the parabola

25. Guided practice Example 1
Identify the following characteristics of the parabola to the right.
(4) x-intercept of the parabola

26. Guided practice Example 1
Identify the following characteristics of the parabola to the right.
(5) domain in interval notation

27. Guided practice Example 1
Identify the following characteristics of the parabola to the right.
(6) range in interval notation

28. Guided practice Example 1
Identify the following characteristics of the parabola to the right.
(7) interval of increase

29. Guided practice Example 1
Identify the following characteristics of the parabola to the right.
(8) interval of decrease

30. Guided practice Example 1
Identify the following characteristics of the parabola to the right.
(9) average rate of change on the interval [0,3]

31. Guided practice Example 1
Identify the following characteristics of the parabola to the right.
(10) circle: maximum value or minimum value

32. Guided practice Example 1
Identify the following characteristics of the parabola to the right.
(11) end behavior

33. INDEPENDENT practice Example 2
Identify the following characteristics of the parabola to the right.
vertex coordinates
equation to the axis of symmetry
y-intercept of the parabola
x-intercept of the parabola
domain
range
interval of increase
interval of decrease
average rat of change on the interval [1,3]
circle: maximum value or minimum value
end behavior

34. Comparing Graphs and Tables of Values of Liner & Quadratic Relations

35. Linear and quadratic relations
A relation is LINEAR if…
A relation is QUADRATIC if…
The graph is a straight line.
The first differences are the same. This
number is the slope of the line.
The equation as degree 1.
The graph is a parabola.
The second differences are equal. If
calculated correctly, the sign of this
number will tell you the direction the parabola
opens.
The equation has degree 2.

36. Watch the entire video. Take Notes.
Mini Lesson #5
Calculating the Second Difference of a
Quadratic Function
I will be able to use the given tables to determine whether the relation is linear, quadratic or neither.
Watch the entire video. Take Notes.

37. INDEPENDENT practice Example 3
Use the given tables to determine whether the relation is linear, quadratic or neither. Show your work.

38. INDEPENDENT practice Example 4
Use the given tables to determine whether the relation is linear, quadratic or neither. Show your work.

39. INDEPENDENT practice Example 5
Use the given tables to determine whether the relation is linear, quadratic or neither. Show your work. X Y 4 3 1 8 -2 13 -5 19 -8 24

40. INDEPENDENT practice Example 6
Use the given tables to determine whether the relation is linear, quadratic or neither. Show your work.

41. Homework
If you need extra room to solve for the intercepts, please attach the additional paper to your homework.
I encourage you to refer to this PowerPoint, re-watch the videos and review the examples if you need additional assistance understanding the concepts.