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Mastering Quadratic Functions: Graphs, Inequalities & Solutions

Learn to graph quadratic functions, solve inequalities, and find roots using the vertex form and quadratic formula. Practice analyzing graphs and identifying key features like the vertex and axis of symmetry.

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Mastering Quadratic Functions: Graphs, Inequalities & Solutions

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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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