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April 2, 2014


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April 2, 2014

  1. 1. Today: STAR Math Update Vertex and Axis of Symmetry: (What they are & how to find them) Graphing Various Quadratic Functions Class Work: 6 Graphs 2April
  2. 2. But first, a sample of Additional Resources Available for download at the v6 math site: Class Notes Section of Notebook, pls. It is strongly recommended that you take good notes today. Also recommended: Bring a calculator everyday BTW, Notebooks will be submitted next week. Get them organized (they should be already)
  3. 3. Textbook & Practice Problems/Quizzes with Answers
  4. 4. Class Zone Class Notes Section
  5. 5. Quadratic Equations vs. Functions Remember, the standard form of a quadratic equation is: ax2 + bx + c = 0 Since the solutions/roots to a standard equation are where the line crosses the x-axis, the y value is always zero at this point. As such, we can substitute y for zero: y = ax2 + bx + c Since the y variable is dependent on the x, or is a function of x, we can substitute the y for the function of x, or (f)x: (f)x = ax2 + bx + c Regardless of which form is presented, the problem is solved in the same way. ***Quadratic Equations are solved algebraically. Quadratic Functions are solved graphically.
  6. 6. 1. To solve and graph a quadratic equation, we need to know where the graph either touches or crosses the x and y axis: These, of course, are the intercepts. In order to graph a quadratic function, we must know to use the equation to plot the key parts of the parabola. Then, we basically connect the dots to complete the graph. Here are those key pieces and how to find them. 1. We will learn a number of ways to find the x-intercepts, but for now we find them by factoring the quadratic equation in standard form. Graphing Parabolas & Parabola Terminology The solutions are the x-intercept(s)
  7. 7. 2.Axis of Symmetry:The axis of symmetry is the verticle or horizontal line which runs through the exact center of the parabola. Graphing Parabolas & Parabola Terminology Other Important points on a Parabola: Another helpful point to remember about the axis of symmetry is that is is always halfway between two x-intercepts
  8. 8. 3. Vertex: The vertex is the highest point (the maximum), or the lowest point (the minimum) on a parabola. Notice that the axis of symmetry always runs through the vertex. Graphing Parabolas & Parabola Terminology If the value of a is negative, the parabola will open downward, and the vertex will be a vertex maximum
  9. 9. Vertex Minimums and Maximums What do the vertex minimum or maximum tell us in terms of the function's domain and range. The information regards the range of the function: No y value can be greater than the vertex maximum, nor less than the vertex minimum.
  10. 10. Finding the Axis of Symmetry & Vertex The center of the parabola crosses the x axis at -6. Since the axis of symmetry always runs through the vertex, the x coordinate for the vertex is -6 also. The formula for finding the axis of symmetry x = - b/2a Our quadratic function is: y = x2 + 12x + 32 But, we still don't know where the vertex lies on the vertical (y) axis.
  11. 11. To find the y-coordinate of the vertex, substitute the value of the x-coordinate back into the equation and find y. Finding the Axis of Symmetry and Vertex y = -62 + 12(-6) + 32. y = 36 - 72 + 32. y = -4 The bottom of the parabola (the vertex) is at -6 on the x axis, and -4 on the y axis. Remember, the axis of symmetry always goes through the vertex; the AOS and the vertex are the same point.
  12. 12. Finding the Axis of Symmetry and Vertex Find the x-intercepts, axis of symmetry & vertex for the following: x2 + 2x – 3 = 0 –2x2 + 6x + 56 = 0 2x2 + 2x = 3 Lastly, Solve a quadratic equation to find the value of x f(x) = -x2 -4x - 12 Today's Assignment: Graph equations, paying special attention to how the a, b, and c values change the shape of the parabola
  13. 13. 2. The Microhard Corporation has found that the equation P = x2 - 7x - 94 describes the profit P, in thousands of dollars, for every x hundred computers sold. How many computers were sold if the profit was $50,000? f(x) = x2 + 2x + 8 f(x) = 2x2 + 4x + 2