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Quadratic Functions

Quadratic Functions

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Quadratic Functions

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  1. Quadratic Functions

  2. Ticket In The Door

  3. Lesson Essential Question What are the important parts of a quadratic graph?

  4. Quadratic Review • For each quadratic function: • Identify the quadratic term (a) • Identify the linear term (b) • Identify the constant term (c)

  5. Quadratic Function: y = ax2 + bx + c • Example 1: 2x2 + 3x + 10 a = _____ b = _____ c = _____ • Example 2: -3x2 + 5x a = _____ b = _____ c = _____ • Example 3: x2 - 8x + 7 a = _____ b = _____ c = _____ • Example 4: -x2 - 9x – 3 a = _____ b = _____ c = _____ • Example 5: -x2 - 6x a = _____ b = _____ c = _____ • Example 6: x2 a = _____ b = _____ c = _____

  6. Consider the following quadratic function: f(x) = x2 + 2x – 3 Let’s talk about another important part of a quadratic function: Where is the y-intercept? Where does the function cross the y-axis? y-intercept: (0, -3)

  7. Consider the following quadratic function: f(x) = x2 + 2x – 3 Let’s talk about another important part of a quadratic function: Where are the x-intercepts? Where does the function cross the x-axis? x-intercepts: (1, 0) & (-3, 0)

  8. Consider the following quadratic function: f(x) = x2 + 2x – 3 Let’s talk about several important parts of a quadratic function: Where is the vertex? (-1, -4)

  9. Consider the following quadratic function: f(x) = x2 + 2x – 3 Let’s talk about another important part of a quadratic function: How do we algebraically calculate the vertex?

  10. Consider the following quadratic function: f(x) = x2 + 2x – 3 Calculating the vertex. The vertex is a coordinate point (x, y) on the graph, now that we have the x value how do you think we determine the y value?

  11. Consider the following quadratic function: f(x) = x2 + 2x – 3 Calculating the vertex. Substitute the value of x into the given function equation above and solve! The answer is the value for y. When x = -1, y = -4. Vertex is: (-1, -4).

  12. Consider the following quadratic function: f(x) = x2 + 2x – 3 Let’s talk about another important part of a quadratic function: What is the axis of symmetry? Now that you see what it is, how would you define the axis of symmetry?

  13. Consider the following quadratic function: f(x) = x2 + 2x – 3 Let’s talk about another important part of a quadratic function: How do we represent this axis of symmetry? x = -1

  14. Consider the following quadratic function: f(x) = x2 – 2x – 15 Where are the x-intercepts? Where does the function cross the x-axis? x-intercepts: (-3, 0) & (5, 0)

  15. Consider the following quadratic function: f(x) = x2 – 2x – 15 Where is the y-intercept? Where does the function cross the y-axis? y-intercept: (0, -15)

  16. Let’s Do It Again Ourselves!!Consider the following quadratic function: f(x) = x2 – 2x – 15 Where is the vertex? Algebraically calculate the vertex. (1, -16)

  17. Consider the following quadratic function: f(x) = x2 – 2x – 15 Where is the axis of symmetry? Draw in the axis of symmetry. What is the axis of symmetry?

  18. Consider the following quadratic function: f(x) = x2 + 3x Where is the y-intercept? Where does the function cross the y-axis? y-intercept: (0, 0)

  19. Consider the following quadratic function: f(x) = x2 + 3x Where are the x-intercepts? Where does the function cross the x-axis? x-intercepts: (-3, 0) & (0, 0)

  20. Let’s Do It Again Ourselves!!Consider the following quadratic function: f(x) = x2 + 3x Where is the vertex? Algebraically calculate the vertex. (-1.5, -2.25)

  21. Consider the following quadratic function: f(x) = x2 + 3x Where is the axis of symmetry? Draw in the axis of symmetry. What is the axis of symmetry?

  22. Now, Visualize the graph! Given: f(x) = x2 – 4x + 3 Open up or down? Calculate the vertex? What is the axis of symmetry? Where is the y-intercept?

  23. Now, Visualize the graph! Given: f(x) = 2x2 + 3x – 1 Open up or down? Calculate the vertex? What is the axis of symmetry? Where is the y-intercept?

  24. Now, Visualize the graph! Given: f(x) = 5x2 – 2x + 5 Open up or down? Calculate the vertex? What is the axis of symmetry? Where is the y-intercept?

  25. Now, Visualize the graph! Given: f(x) = x2 – 2x – 15 Open up or down? Calculate the vertex? What is the axis of symmetry? Where is the y-intercept?

  26. Ticket Out The Door • Complete the ticket out the door problem. Please hand it to me as you walk out of the door. Homework • Complete the worksheet for homework.

  27. IMPORTANT PARTS OF QUADRATIC GRAPHS • Does the graph open up or down • (write “a” is + or -) • Put a star at the Vertex (write the point) • Draw the Axis of Symmetry and write the equation • Circle the X-intercepts (write the point) • Draw a square around the Y-intercept (write the point)

  28. Quadratic Functions and their important parts! What important parts do you recognize in this graph? y = x2 – 3x – 10

  29. Quadratic Functions and their important parts! What important parts do you recognize in this graph?

  30. Lesson Essential Question How do you graph a quadratic function using the vertex?

  31. Putting It All Together Now!!! Graphing Parabolas In order to graph we will need the following: • Visualize whether the parabola open up or down • Calculate the coordinates of the Vertex • Determine the Axis of Symmetry • Determine the y-intercept • Plot a few more points to understand the actual shape of the graph • Identify the x-intercepts

  32. Calculate the vertex and identify the axis of symmetry (AOS).

  33. Graphing Quadratic Functions Graph the function, then identify the x-intercepts (roots) = ____________

  34. Graphing Quadratic Functions Graph the function, then identify the x-intercepts (roots) = ____________

  35. Graphing Quadratic Functions Graph the function, then identify the x-intercepts (roots) = ____________

  36. Graphing Quadratic Functions Graph the function, then identify the x-intercepts (roots) = ____________

  37. Graphing Quadratic Functions e.) Sketch the graph of y = x2 – 2x – 3 Graph the function, then identify the x-intercepts (roots) = ____________

  38. Graphing Quadratic Functions f.) Sketch the graph of y = x2 + 4x + 4 Graph the function, then identify the x-intercepts (roots) = ____________

  39. Graphing Quadratic Functions g.) Sketch the graph of y = ½x2 – 3 Graph the function, then identify the x-intercepts (roots) = ____________

  40. Graphing Quadratic Functions h.) Sketch the graph of y = 2x2 + 4x + 5 Graph the function, then identify the x-intercepts (roots) = ____________

  41. On Your Own Practice Please complete the practice worksheets in order to develop and master this skill. Thank you 

  42. Homework Assignment More Practice Graphing Quadratic Functions!