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Understanding Quadratic Functions: Key Features and Characteristics

Learn to interpret quadratic functions and identify key features like vertex, axis of symmetry, intercepts, and intervals of increase and decrease with real-world examples of parabolic antennas. Practice identifying domain, range, and end behavior of quadratic functions.

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Understanding Quadratic Functions: Key Features and Characteristics

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  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 wheny = 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 #1Identifying 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. Parenthesesand/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) • https://www.youtube.com/watch?v=k7oAl3AiV2I • Video – Take Notes - (7:30-8:30) • https://www.youtube.com/watch?v=k7oAl3AiV2I

  8. Mini Lesson #1Identifying 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) • https://www.youtube.com/watch?v=k7oAl3AiV2I

  9. Mini Lesson #2Finding the end behavior of a quadratic function • The end behavior of a function is thebehavior 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 theend behavior of the graph. Watch the entire video below. Take Notes. • https://www.youtube.com/watch?v=wXSrcEMcbeo 2nd Degree & Negative Leading Coefficient 2nd Degree & Positive Leading Coefficient

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

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

  12. Mini Lesson #3identify the increasing and decreasing intervals • Video – Take Notes - (2:47-4:35) • https://www.youtube.com/watch?v=mkG0R5P4gZk

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

  16. Mini Lesson #3identify 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 #4identify the intercepts of a quadratic function • Video – Take Notes - (0:00 - 2:35) • https://www.youtube.com/watch?v=MqIpwRD-bLA

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

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

  20. Guided practiceExample 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

  21. Guided practiceExample 1 Identify the following characteristics of the parabola to the right. • vertex coordinates

  22. Guided practiceExample 1 Identify the following characteristics of the parabola to the right. • (2) equation to the axis of symmetry

  23. Guided practiceExample 1 Identify the following characteristics of the parabola to the right. • (3) y-intercept of the parabola

  24. Guided practiceExample 1 Identify the following characteristics of the parabola to the right. • (4) x-intercept of the parabola

  25. Guided practiceExample 1 Identify the following characteristics of the parabola to the right. • (5) domain in interval notation

  26. Guided practiceExample 1 Identify the following characteristics of the parabola to the right. • (6) range in interval notation

  27. Guided practiceExample 1 Identify the following characteristics of the parabola to the right. • (7) interval of increase

  28. Guided practiceExample 1 Identify the following characteristics of the parabola to the right. • (8) interval of decrease

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

  30. Guided practiceExample 1 Identify the following characteristics of the parabola to the right. • (10) circle: maximum value or minimum value

  31. Guided practiceExample 1 Identify the following characteristics of the parabola to the right. • (11) end behavior

  32. INDEPENDENT practiceExample 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

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

  34. Linear and quadratic relations A relation is LINEAR if… A relation is QUADRATIC if… 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. • 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.

  35. 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. https://www.youtube.com/watch?v=OCRoD0jZF7o

  36. INDEPENDENT practiceExample 3 • Use the given tables to determine whether the relation is linear, quadratic or neither. Show your work.

  37. INDEPENDENT practiceExample 4 • Use the given tables to determine whether the relation is linear, quadratic or neither. Show your work.

  38. INDEPENDENT practiceExample 5 • Use the given tables to determine whether the relation is linear, quadratic or neither. Show your work.

  39. INDEPENDENT practiceExample 6 • Use the given tables to determine whether the relation is linear, quadratic or neither. Show your work.

  40. 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.

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