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Introduction

Introduction

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Introduction

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  1. Introduction Imagine the path of a basketball as it leaves a player’s hand and swooshes through the net. Or, imagine the path of an Olympic diver as she arcs away from the diving board, slips deep into the water, and resurfaces. Both traveling paths make a U-shaped curve known as a parabola. A parabola is the U-shaped graph of a quadratic function, which is an equation with a degree of 2. Formally, a quadratic function is a function that can be written in the form f(x) = ax2 + bx+ c, where a ≠ 0. Quadratics can be used to model varying situations. In this lesson, you will learn about and practice identifying the important points on a parabola and use those points to sketch the corresponding graph. 5.3.1: Creating and Graphing Equations Using Standard Form

  2. Key Concepts The standard form of a quadratic function is f(x) = ax2 + bx + c, where a is the coefficient of the quadratic term, b is the coefficient of the linear term, and c is the constant term. The x-intercepts of the graph of a quadratic function are the points at which the graph crosses the x-axis, and are written as (x, 0). The y-intercept of the graph of a quadratic function is the point at which the graph crosses the y-axis and is written as (0, y). The graph on the following slide shows the location of the parabola’s y-intercept. 5.3.1: Creating and Graphing Equations Using Standard Form

  3. Key Concepts, continued 5.3.1: Creating and Graphing Equations Using Standard Form

  4. Key Concepts, continued The equation of the parabola is y = x2 – 2x – 4. Note that the y-intercept of this equation is –4 and the constant term of the quadratic equation is –4. The y-intercept of a quadratic is the c value of the quadratic equation when written in standard form. The axis of symmetry of a parabola is the line through the vertex of a parabola about which the parabola is symmetric. 5.3.1: Creating and Graphing Equations Using Standard Form

  5. Key Concepts, continued 5.3.1: Creating and Graphing Equations Using Standard Form

  6. Key Concepts, continued The axis of symmetry extends through the vertex of the graph. The vertex of a parabola is the point on a parabola where the graph changes direction, (h, k), where h is the x-coordinate and k is the y-coordinate. The equation of the axis of symmetry is In the example above, the equation of the axis of symmetry is x = 1 because the vertical line through 1 is the line that cuts the parabola in half. 5.3.1: Creating and Graphing Equations Using Standard Form

  7. Key Concepts, continued The axis of symmetry is sometimes written as where h is the x-coordinate of the vertex. The vertex of a quadratic lies on the axis of symmetry, as shown in the graph on the next slide. The formula is not only used to find the equation of the axis of symmetry, but also to find the x-coordinate of the vertex. 5.3.1: Creating and Graphing Equations Using Standard Form

  8. Key Concepts, continued 5.3.1: Creating and Graphing Equations Using Standard Form

  9. Key Concepts, continued To find the y-coordinate, substitute the value of x into the original function, The vertex gives information about the maximum or minimum value of a quadratic. The maximum is the largest y-value of a quadratic and the minimum is the smallest y-value. One might see these terms in real-world situations such as maximizing profit and minimizing cost, among others. 5.3.1: Creating and Graphing Equations Using Standard Form

  10. Key Concepts, continued From the equation of a function in standard form, you can determine if you have a maximum or minimum based on the sign of the coefficient of the quadratic term, a. If a > 0, the parabola opens up and therefore has a minimum value. If a < 0, the parabola opens down and therefore has a maximum value. 5.3.1: Creating and Graphing Equations Using Standard Form

  11. Key Concepts, continued 5.3.1: Creating and Graphing Equations Using Standard Form

  12. Key Concepts, continued To create the equation of a quadratic in standard form, one must know some combination of the following key features: y-intercept, axis of symmetry or the vertex, and maximum or minimum. The key features of a quadratic are the x-intercepts, y-intercept, where the function is increasing and decreasing, where the function is positive and negative, relative minimums and maximums, symmetries, and end behavior of the function. 5.3.1: Creating and Graphing Equations Using Standard Form

  13. Key Concepts, continued To create the equation of a quadratic given the vertex and y-intercept, first set the x-coordinate of the vertex equal to and isolate b. Second, substitute the expression that is equal to b, the y-intercept, and the coordinates of the vertex into the standard form of a quadratic equation, y = ax2 + bx+ c. Solve the equation for a and then substitute the value of a into the equation you created in the first step to determine the value of b. 5.3.1: Creating and Graphing Equations Using Standard Form

  14. Common Errors/Misconceptions forgetting to write an equation for the axis of symmetry not finding and labeling all important points drawing the parabola as a straight line or a series of straight lines instead of a curve 5.3.1: Creating and Graphing Equations Using Standard Form

  15. Guided Practice Example 2 h(x) = 2x2 – 11x + 5 is a quadratic function. Determine the direction in which the function opens, the vertex, the axis of symmetry, the x-intercept(s), and the y-intercept. Use this information to sketch the graph. 5.3.1: Creating and Graphing Equations Using Standard Form

  16. Guided Practice: Example 2, continued Determine whether the graph opens up or down. h(x) = 2x2 – 11x + 5 is in standard form; therefore, a = 2. Since a > 0, the parabola opens up. 5.3.1: Creating and Graphing Equations Using Standard Form

  17. Guided Practice: Example 2, continued Find the vertex and the equation of the axis of symmetry. h(x) = 2x2 – 11x + 5 is in standard form; therefore, a = 2and b = –11. The vertex has an x-value of 2.75. 5.3.1: Creating and Graphing Equations Using Standard Form

  18. Guided Practice: Example 2, continued Since the input value is 2.75, find the output value by evaluating the function for h(2.75). The y-value of the vertex is –10.125. The vertex is the point (2.75, –10.125). Since the axis of symmetry is the vertical line through the vertex, the equation of the axis of symmetry is x = 2.75. 5.3.1: Creating and Graphing Equations Using Standard Form

  19. Guided Practice: Example 2, continued Find the y-intercept. h(x) = 2x2 – 11x + 5 is in standard form, so the y-intercept is the constant c, which is 5. The y-intercept is (0, 5). 5.3.1: Creating and Graphing Equations Using Standard Form

  20. Guided Practice: Example 2, continued Find the x-intercepts, if any exist. The x-intercepts occur when y = 0. Substitute 0 for the output, h(x), and solve. This equation is factorable, but if we cannot easily identify the factors, the quadratic formula always works. Note both methods shown on the following slides. 5.3.1: Creating and Graphing Equations Using Standard Form

  21. Guided Practice: Example 2, continued Solved by factoring: h(x) = 2x2 – 11x + 5 0 = 2x2 – 11x + 5 0 = (2x – 1)(x – 5) 0 = 2x – 1 or 0 = x – 5 x = 0.5 or x = 5 Solved using the quadratic formula: h(x) = 2x2 – 11x + 5 0 = 2x2 – 11x + 5 a = 2, b = –11, and c = 5 5.3.1: Creating and Graphing Equations Using Standard Form

  22. Guided Practice: Example 2, continued 5.3.1: Creating and Graphing Equations Using Standard Form

  23. Guided Practice: Example 2, continued The x-intercepts are (0.5, 0) and (5, 0). 5.3.1: Creating and Graphing Equations Using Standard Form

  24. Guided Practice: Example 2, continued Plot the points from steps 2–4 and their symmetric points over the axis of symmetry. Connect the points with a smooth curve. 5.3.1: Creating and Graphing Equations Using Standard Form

  25. Guided Practice: Example 2, continued ✔ 5.3.1: Creating and Graphing Equations Using Standard Form

  26. Guided Practice: Example 2, continued 5.3.1: Creating and Graphing Equations Using Standard Form

  27. Guided Practice Example 5 Create the equation of a quadratic function given a vertex of (2, –4) and a y-intercept of 4. 5.3.1: Creating and Graphing Equations Using Standard Form

  28. Guided Practice: Example 5, continued Write an equation for b in terms of a. Set the x-coordinate of the vertex equal to 5.3.1: Creating and Graphing Equations Using Standard Form

  29. Guided Practice: Example 5, continued Substitute the expression for b from step 1, the coordinates of the vertex, and the y-intercept into the standard form of a quadratic equation. 5.3.1: Creating and Graphing Equations Using Standard Form

  30. Guided Practice: Example 5, continued 5.3.1: Creating and Graphing Equations Using Standard Form

  31. Guided Practice: Example 5, continued Substitute the value of a into the equation for b from step 1. 5.3.1: Creating and Graphing Equations Using Standard Form

  32. Guided Practice: Example 5, continued Substitute a, b, and c into the standard form of a quadratic equation. The equation of the quadratic function with a vertex of (2, –4) and a y-intercept of 4 is y = 2x2– 8x + 4. ✔ 5.3.1: Creating and Graphing Equations Using Standard Form

  33. Guided Practice: Example 5, continued 5.3.1: Creating and Graphing Equations Using Standard Form

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