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Exploring Quadratics and Polynomial Functions for Better Understanding

Learn about quadratic equations, nature of roots, discriminants, polynomial functions, graphing, and behavior analysis to deepen your mathematical knowledge. Utilize the quadratic formula, discriminant, and leading coefficient tests to interpret various functions.

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Exploring Quadratics and Polynomial Functions for Better Understanding

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  1. Solving Quadratics Quadratic Formula Discriminant – Nature of the Roots

  2. Warm Up – Question 1 • Given • and • find an expression for the composite function and state its domain restrictions.

  3. Warm Up Answer

  4. Warm Up – Question 2 • Given a) b)

  5. Solve by Quadratic Formula

  6. Derive the Quadratic Formulaby Completing the Square

  7. Solve using the Quadratic Formula

  8. Solve using the Quadratic Formula

  9. Solve using the Quadratic Formula

  10. Solve using the Quadratic Formula

  11. Solve using the Quadratic Formula

  12. Solve using the Quadratic Formula

  13. Solve using the Quadratic Formula

  14. Solve using the Quadratic Formula

  15. The Discriminant The nature of the roots

  16. The Discriminant • It comes from the quadratic formula. • =

  17. When you apply the quadratic formula to any quadratic equation, you will find that the value of b²-4ac is either positive, negative, or 0. • The value of the discriminant is what tells us the nature of the roots (solutions) to the quadratic.

  18. Solutions of a Quadratic Equation • If

  19. Real Numbers (ℝ) Rational Numbers (ℚ) Irrational Numbers Integers (ℤ) Decimal form is non-terminating and non-repeating Whole Numbers Natural Numbers (ℕ) 1, 2, 3, … 0, 1, 2, 3, … …-3, -2, -1, 0, 1, 2, 3, … Decimal form either terminates or repeats All rational and irrational numbers

  20. Nature of the Roots Find the Discriminant and Describe its Roots.

  21. Nature of the Roots Find the Discriminant and Describe its Roots.

  22. Nature of the Roots Find the Discriminant and Describe its Roots

  23. Graphs of Polynomial Functions

  24. Explore – Look at the relationship between the degree & sign of the leading coefficient and the right- and left-hand behavior of the graph of the function.

  25. Explore – Look at the relationship between the degree & sign of the leading coefficient and the right- and left-hand behavior of the graph of the function.

  26. Explore – Look at the relationship between the degree & sign of the leading coefficient and the right- and left-hand behavior of the graph of the function.

  27. Continuous Function • A function is continuous if its graph can be drawn with a pencil without lifting the pencil from the paper. Continuous Not Continuous

  28. Polynomial Function • Polynomial Functions have continuous graphs with smooth rounded turns. • Written: • Example:

  29. Explore using graphing CalculatorDescribe graph as S or W shaped.

  30. Generalizations? • The number of turns is one less than the degree. • Even degree → “W” Shape • Odd degree → “S” Shape

  31. Describe the Shape and Number of Turns.

  32. Let’s explore some more….we might need to revise our generalization. • Take a look at the following graph and tell me if your conjecture is correct.

  33. Lead Coefficient Test When n is odd Lead Coefficient is Positive: (an >0), the graph falls to the left and rises to the right Lead Coefficient is Negative: (an <0), the graph rises to the left and falls to the right

  34. Lead Coefficient Test When n is even Lead Coefficient is Positive: (an >0), the graph rises to the left and rises to the right Lead Coefficient is Negative: (an <0), the graph falls to the left and falls to the right

  35. Leading Coefficient: an a > 0 a < 0

  36. Use the Leading Coeffiicent Test to describe the right-hand and left-hand behavior of the graph of each polynomial function:

  37. Use the Leading Coeffiicent Test to describe the right-hand and left-hand behavior of the graph of each polynomial function:

  38. Use the Leading Coeffiicent Test to describe the right-hand and left-hand behavior of the graph of each polynomial function:

  39. Use the Leading Coeffiicent Test to describe the right-hand and left-hand behavior of the graph of each polynomial function:

  40. A polynomial function (f) of degree n , the following are true • The function has at most n real zeros • The graph has at most (n-1) relative extrema (relative max/min)

  41. Local Max / Min (in terms of y)Increasing / Decreasing (in terms of x)

  42. Local Max / Min (in terms of y)Increasing / Decreasing (in terms of x)

  43. Approximate any local maxima or minima to the nearest tenth.Find the intervals over which the function is increasing and decreasing.

  44. Find the Zeros of the polynomial function below and sketch on the graph:

  45. Find the Zeros of the polynomial function below and sketch on the graph: Multiplicity of 2 – EVEN - Touches

  46. Find the Zeros of the polynomial function below and sketch on the graph:

  47. Find the Zeros of the polynomial function below and sketch on the graph: NO X-INTERCEPTS!

  48. Find the Zeros of the polynomial function below and sketch on the graph:

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