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Quadratic Functions - Graphing, Solving, and Inequalities

This unit covers quadratic functions, including graphing, finding the vertex, solving by factoring and quadratic formula, and solving quadratic inequalities.

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Quadratic Functions - Graphing, Solving, and Inequalities

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  1. Unit 5Quadratics

  2. Quadratic Functions • Any function that can be written in the form

  3. Quadratic Functions • Graph forms a parabola • Label the parts of the parabola or

  4. To find the axis of symmetry • When

  5. Find the vertex and los

  6. Vertex (h,k) form of a Quadratic • Standard Form:

  7. Parent Function

  8. Transformations • You can tell what the graph of the quadratic will look like if the eq. is in (h,k) form

  9. Sketch the graph

  10. Sketch the graph

  11. Sketch the graph

  12. Sketch the graph

  13. Sketch the graph

  14. Identifying Important Parts on Calculator • 2nd calc—then select max or min

  15. Completing the Square • Used to go from standard form to (h,k) form or to get the equation in the form of a perfect square to solve • Steps: • Move the constant • Factor out the # in front of x2 • Take ½ of middle term and square it • Write in factored form for the perfect sq. trinomial • Add to both sides (multiply by # in front) • Move constant back to get in (h,k) form

  16. Examples

  17. Examples

  18. Example

  19. Example

  20. Example

  21. Solving Quadratics • You can solve by graphing, factoring, square root method, and quadratic formula • Solutions, roots, or zeros

  22. Solving by Graphing • Graph the parabola • Look for where is crosses the x-axis (where y=0) • May have 2 real, 1 real, or no real solutions (Show on calculator) Review finding the vertex

  23. Solve the following by graphing

  24. Solving Quadratics by Factoring • Factor the quadratic • Set each factor that contains a variable equal to zero and solve (zero product property)

  25. More solving by factoring

  26. You Try

  27. Writing the Quadratic Eq. • Write the quadratic with the given roots of ½ and -5

  28. Write the quadratic with • Roots of 2/3 and -2

  29. When solving • Graphing—not always best unless you have exact answers • Factoring—not every polynomial can be factored • Quadratic Formula—always works • Square Root method—may have to complete the square first

  30. Solving using Quadratic formula • Must be in standard form • Identify a, b, and c

  31. Examples

  32. Examples

  33. Examples

  34. Examples

  35. Discriminant • Used to identify the “type” of solutions you will have (without having to solve)

  36. If the discriminant is… • A perfect square---2 rational solutions • A non perfect sq—2 irrational sol. • Zero—1 rational sol. • Negative—2 complex sol.

  37. Identify the nature of the solution

  38. Solving Quadratics using the Sq. Rt. method • Useful when you have x2 = constant or a perfect sq. trinomial ex. (x-3)2=constant • Get the x2 by itself • Take the square rt. of both sides • Don’t forget + or – in your answer!!!

  39. Examples

  40. Examples

  41. Examples

  42. Quadratic Inequalities • Graphing quadratic inequalities in 2 variables: • Steps: • Graph the related equation • Test a point not on the graph of the parabola • Shade region that contains the point if it makes the inequality true or shade the other region if it does not make the inequality true • Ex. Ex.

  43. Graphing Quadratic Inequalities

  44. Solving Quadratic Inequalities • Solving Quadratic Inequalities in one variable: May be solved by graphing or algebraically. • To solve by graphing: • Steps: • Put the inequality in standard form • Find the zeros and sketch the graph of the related equation • identify the x values for which the graph lies below the x-axis if the inequality sign is < or • identify the x values for which the graph lies above the x-axis if the inequality sign is > or

  45. Solve by graphing Solutions:_______________________ Solutions:_______________________

  46. To solve algebraically: • Steps: • Solve the related equation • Plot the zeros on a number line—decide whether or not the zeros are actually included in the solution set • Test all regions of the number to determine other values to include in the solution set

  47. Solve Algebraically

  48. Solving Quadratic Inequalities

  49. Word Problems

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