392 CHAPTER 6 Algebra, Graphs, and Functions MATHEMATICS TODAY Computing Reality with Mathematical Models It is a relatively easy matter for scientists and mathematicians to describe and predict a simple motion like that of a falling object. When the phenomenon is complicated, such as making an accurate prediction of the weather, the mathematics becomes much more difficult. The National Weather Service has devised an algorithm that takes the temperature, pressure, moisture content, and wind velocity of more than 250,000 points in Earth’s atmosphere and applies a set of equations that, it believes, will reasonably predict what will happen at each point over time. Researchers at the National Center for Super-computing Applications are working on a computer model that simulates thunderstorms. They want to know why some thunderstorms can turn severe, even deadly. Why This Is Important Computers are used to model real-life situations, including weather forecasting, travel time estimations, and stock market predictions. Example 6 Describing the Graph of a Quadratic Equation Consider the equation = − + − y x x 3 12 5. 2 a) Determine whether the graph of the equation will be a parabola that opens upward or downward. b) Determine the equation of the axis of symmetry of the parabola. c) Determine the vertex of the parabola. Solution a) Since = − a 3, which is less than 0, the parabola opens downward. b) To determine the equation of the axis of symmetry, we use the equation x . b a2 = − In the equation = − + − = − = = − y x x a b c 3 12 5, 3, 12, and 5, so 2 = − = − − = − − = x b a2 (12) 2( 3) 12 6 2 The equation of the axis of symmetry is = x 2. c) The x-coordinate of the vertex is 2 from part (b). To determine the y-coordinate, we substitute 2 for x in the equation = − + − y x x 3 12 5 2 and then evaluate. = − + − = − + − = − + − = − + − = y x x 3 12 5 3(2) 12(2) 5 3(4) 24 5 12 24 5 7 2 2 Therefore, the vertex of the parabola is located at the point (2, 7) on the graph. 7 Now try Exercise 49 GRAPHING A QUADRATIC EQUATION 1. Determine whether the parabola opens upward or downward. 2. Determine the equation of the axis of symmetry. 3. Determine the vertex of the parabola. 4. Determine the y -intercept by substituting = x 0 into the equation. 5. Determine the x-intercepts (if they exist) by substituting = y 0 into the equation and solving for x. 6. Draw the graph, making use of the information gained in Steps 1 through 5. Remember that the parabola will be symmetric with respect to the axis of symmetry. PROCEDURE In Step 5, you may use either factoring or the quadratic formula to determine the x-intercepts. Example 7 Graphing a Quadratic Equation Graph the equation = − + y x x6 8. 2 Solution We follow the steps outlined in the general procedure. 1. Since = a 1, which is greater than 0, the parabola opens upward. 2. Axis of symmetry: = − = − − = = x b a2 ( 6) 2(1) 6 2 3 Thus, the equation of the axis of symmetry is = x 3. Gorodenkoff/Shutterstock
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