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Chapter 5

Chapter 5. Quadratic Functions & Inequalities. 5.1 – 5.2 Graphing Quadratic Functions. The graph of any Quadratic Function is a Parabola To graph a quadratic Function always find the following: y-intercept (c - write as an ordered pair) equation of the axis of symmetry x =

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Chapter 5

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  1. Chapter 5 Quadratic Functions & Inequalities

  2. 5.1 – 5.2 Graphing Quadratic Functions • The graph of any Quadratic Function is a Parabola • To graph a quadratic Function always find the following: • y-intercept (c - write as an ordered pair) • equation of the axis of symmetry x = • vertex- x and y values (use x value from AOS and solve for y) • roots (factor) These are the solutions to the quadratic function • minimum or maximum • domain and range If a is positive = opens up (minimum) – y coordinate of the vertex If a is negative = opens down (maximum) – y coordinate of the vertex

  3. Ex: 1 Graph by using the vertex, AOS and a table • f(x) = x2 + 2x - 3

  4. Graph Find the y-int, AOS, vertex, roots, minimum/maximum, and domain and range • f(x) = -x2 + 7x – 14

  5. Graph Find the y-int, AOS, vertex, roots, minimum/maximum, and domain and range • f(x) = 4x2 + 2x - 3

  6. Graph Find the y-int, AOS, vertex, roots, minimum/maximum, and domain and range • x2 + 4x + 6 = f(x)

  7. Graph Find the y-int, AOS, vertex, roots, minimum/maximum, and domain and range • 2x2 – 7x + 5 = f(x)

  8. 5.7 Analyzing graphs of Quadratic Functions • Most basic quadratic function is • y = x2 • Axis of Symmetry is x = 0 • Vertex is (0, 0) • A family of graphs is a group of graphs that displays one or more similar characteristics! • y = x2 is called a parent graph

  9. Vertex Form y = a(x – h)2 + k • Vertex: (h, k) • Axis of symmetry: x = h • a is positive: opens up, a is negative: opens down • Narrower than y = x2 if |a| > 1, Wider than y = x2 if |a| < 1 • h moves graph left and right • - h moves right • + h moves left • k moves graph up or down • - k moves down • + k moves up

  10. Identify the vertex, AOS, and direction of opening. State whether it will be narrower or wider than the parent graph • y = -6(x + 2)2 – 1 • y = (x - 3)2 + 5 • y = 6(x - 1)2 – 4 • y = - (x + 7)2

  11. Graph after identifying the vertex, AOS, and direction of opening. Make a table to find additional points. y = 4(x+3)2 + 1

  12. Graph after identifying the vertex, AOS, and direction of opening. Make a table to find additional points. y = -(x - 5)2 – 3

  13. Graph after identifying the vertex, AOS, and direction of opening. Make a table to find additional points. y = ¼ (x - 2)2 + 4

  14. 5.8 Graphing and Solving Quadratic Inequalities • 1. Graph the quadratic equation as before (remember dotted or solid lines) • 2. Test a point inside the parabola • 3. If the point is a solution(true) then shade the area inside the parabola if it is not (false) then shade the outside of the parabola

  15. Example 1: Graph y > x2 – 10x + 25

  16. Example 2: Graph y < x2 - 16

  17. Example 3: Graph y < -x2 + 5x + 6

  18. Example 4: Graph y > x2 – 3x + 2

  19. 5.4 Complex Numbers • Let’s see… Can you find the square root of a number? A. B. C. D. E. F. G.

  20. So What’s new? • To find the square root of negative numbers you need to use imaginary numbers. • i is the imaginary unit • i2 = -1 • i = Square Root Property For any real number x, if x2 = n, then x = ±

  21. What about the square root of a negative number? A. C. B. D. E.

  22. Let’s Practice With i • Simplify -2i (7i) (2 – 2i) + (3 + 5i) i45 i31 A. B. C. D. E.

  23. Solve 3x2 + 48 = 0 4x2 + 100 = 0 x2 + 4= 0 A. B. C.

  24. 5.4 Day #2More with Complex Numbers • Multiply • (3 + 4i) (3 – 4i) • (1 – 4i) (2 + i) • (1 + 3i) (7 – 5i) • (2 + 6i) (5 – 3i)

  25. *Reminder: You can’t have i in the denominator • Divide 3i 5 + i 2 + 4i 2i -2i 4 - i 3 + 5i 5i 2 + i 1 - i A. D. B. E. C.

  26. 5.5 Completing the Square

  27. 5.5 Completing the Square Let’s try some: Solve:

  28. 5.6 The Quadratic Formula and the Discriminant The discriminant:the expression under the radical sign in the quadratic formula. *Determines what type and number of roots

  29. 5.6 The Quadratic Formula and the Discriminant • The Quadratic Formula: • Use when you cannot factor to find the roots/solutions

  30. Example 1: Use the discriminant to determine the type and number of roots, then find the exact solutions by using the Quadratic Formula • x2 – 3x – 40 = 0

  31. Example 2: Use the discriminant to determine the type and number of roots, then find the exact solutions by using the Quadratic Formula • 2x2 – 8x + 11 = 0

  32. Example 3: Use the discriminant to determine the type and number of roots, then find the exact solutions by using the Quadratic Formula • x2 + 6x – 9 = 0

  33. TOD: Solve using the method of your choice! (factor or Quadratic Formula) A. 7x2 + 3 = 0 B. 2x2 – 5x + 7 = 3 C. 2x2 - 5x – 3 = 0 D. -x2 + 2x + 7 = 0

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