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Chapter 9

Chapter 9. Section 5. More on Graphing Quadratic Equations; Quadratic Functions. Graph quadratic equations of the form y = ax 2 + bx + c ( a  0). Use a graph to determine the number of real solutions of a quadratic equation. 9.5. 2.

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Chapter 9

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  1. Chapter 9 Section 5

  2. More on Graphing Quadratic Equations; Quadratic Functions Graph quadratic equations of the form y = ax2 + bx + c (a 0). Use a graph to determine the number of real solutions of a quadratic equation. 9.5 2

  3. In Section 5.4, we graphed the quadratic equation y = x2. By plotting points, we obtained the graph of a parabola shown here. More on Graphing Quadratic Equations; Quadratic Functions Recall the lowest (or highest point if the parabola opens downward) point on the graph is called the vertexof the parabola. The vertical line through the vertex is called the axis, or axis of symmetry. The two halves of the parabola are mirror images of each other across this axis. Slide 9.5-3

  4. Objective 1 Graph quadratic equations of the form y = ax2 + bx + c (a 0). Slide 9.5-4

  5. Graph quadratic equations of the form y = ax2 + bx + c (a 0). Every equation of the form y = ax2 + bx + c with a 0, has a graph that is a parabola. The vertex is the most important point to locate when graphing a quadratic equation. Earlier in the course we solved linear equations in one variable that were of the form Ax + B = C; then graphed linear equations in two variables that were of the form y = mx + b. Now, we are ready to do the same sort of thing for quadratic equations. We know how to solve ax2 + bx + c = 0; and now we graph y = ax2 + bx + c. Slide 9.5-5

  6. Graphing the Parabolay = ax2 + bx + c Step 1:Find the vertex.Let x = and find the corresponding y-value by substituting for x in the equation. Graph quadratic equations of the form y = ax2 + bx + c (a 0). (cont’d) Step 2:Find the y-intercept. Let x = 0 and solve for y. Step 3:Find the x-intercepts(if they exist). Let y = 0 and solve for x. Step 4:Plotthe intercepts and the vertex. Step 5:Find and plot additional ordered pairsnear the vertex and intercepts as needed, using symmetry about the axis of the parabola. Slide 9.5-6

  7. Graph y = x2 + 2x– 8. EXAMPLE 1 Graphing a Parabola by Finding the Vertex and Intercepts Solution: Slide 9.5-7

  8. Graph y = –x2 + 2x + 4. EXAMPLE 2 Graphing a Parabola Solution: Slide 9.5-8

  9. Objective 2 Use a graph to determine the number of real solutions of a quadratic equation. Slide 9.5-9

  10. Using the vertical line test from Section 3.6, we see that the graph of an equation of the form y = ax2 + bx + c is the graph of a function. A function defined by an equation of the form (x) = ax2 + bx + c (a  0) is called a quadratic function. The domain(possible x-values) of a quadratic function is the set of all real numbers, or (–,); the range (the resulting y-values) can be determined after the function is graphed. Use a graph to determine the number of real solutions of a quadratic equation. Slide 9.5-10

  11. Decide from the graph how many real number solutions the corresponding equation f (x)= 0 has. Give the solution set. EXAMPLE 3 Determining the Number of Real Solutions from Graphs Solution: There are two real solutions, {±2}, that correspond to the graph. Slide 9.5-11

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