Graphing and Solving Quadratic Inequalities

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Presentation transcript:

Graphing and Solving Quadratic Inequalities Algebra II Mr. Gilbert Chapter 6.7 Graphing and Solving Quadratic Inequalities Standard & Honors 11/30/2018

Agenda Warm up Homework Review Lesson New Homework Check your answers 11/30/2018

Click the mouse button or press the Space Bar to display the answers. 11/30/2018 Click the mouse button or press the Space Bar to display the answers. Transparency 7

11/30/2018 Transparency 7a

Homework Review 11/30/2018

6.7 Graphing and solving Quadratic Inequalities Example 1 Graph a Quadratic Inequality (4) Example 2 Solve ax2 + bx + c  0 (3) Example 3 Solve ax2 + bx + c  0 (5) Example 4 Write an Inequality (4) Example 5 Solve a Quadratic Inequality (5) 11/30/2018 Lesson 7 Contents

Graph Step 1 Graph the related quadratic equation, Since the inequality symbol is >, the parabola should be dashed. 11/30/2018 Example 7-1a

Step 2 Test a point inside the parabola, such as (1, 2). Graph Step 2 Test a point inside the parabola, such as (1, 2). (1, 2) So, (1, 2) is a solution of the inequality. 11/30/2018 Example 7-1a

Step 3 Shade the region inside the parabola. Graph Step 3 Shade the region inside the parabola. (1, 2) (1, 2) 11/30/2018 Example 7-1a

Graph Answer: 11/30/2018 Example 7-1b

Solve by graphing. The solution consists of the x values for which the graph of the related quadratic function lies above the x-axis. Begin by finding the roots of the related equation. Related equation Factor. Zero Product Property or Solve each equation. 11/30/2018 Example 7-2a

Answer: The solution set is Sketch the graph of the parabola that has x-intercepts at 3 and 1. The graph lies above the x-axis to the left of and to the right of Answer: The solution set is 11/30/2018 Example 7-2a

Solve by graphing. Answer: 11/30/2018 Example 7-2b

Use the Quadratic Formula. Solve by graphing. This inequality can be rewritten as The solution consists of the x-values for which the graph of the related quadratic equation lies on and above the x-axis. Begin by finding roots of the related equation. Related equation Use the Quadratic Formula. Replace a with –2, b with –6 and c with 1. 11/30/2018 Example 7-3a

Simplify and write as two equations. or Sketch the graph of the parabola that has x-intercepts of –3.16 and 0.16. The graph should open down since a < 0. 11/30/2018 Example 7-3a

Answer: The graph lies on and above the x-axis at and and between these two values. The solution set of the inequality is approximately 11/30/2018 Example 7-3a

Check Test one value of x less than –3. 16, one between –3. 16 and 0 Check Test one value of x less than –3.16, one between –3.16 and 0.16, and one greater than 0.16 in the original inequality. 11/30/2018 Example 7-3a

Solve by by graphing. Answer: 11/30/2018 Example 7-3b

Subtract 15 from each side. Sports The height of a ball above the ground after it is thrown upwards at 40 feet per second can be modeled by the function where the height h(x) is given in feet and the time x is in seconds. At what time in its flight is the ball within 15 feet of the ground? The function h(x) describes the height of the ball. Therefore, you want to find values of x for which Original inequality Subtract 15 from each side. 11/30/2018 Example 7-4a

Graph the related function using a graphing calculator. The zeros are about 0.46 and 2.04. The graph lies below the x-axis when or 11/30/2018 Example 7-4a

Answer: Thus, the ball is within 15 feet of the ground for the first 0 Answer: Thus, the ball is within 15 feet of the ground for the first 0.46 second of its flight and again after 2.04 seconds until the ball hits the ground at 2.5 seconds. 11/30/2018 Example 7-4a

Sports The height of a ball above the ground after it is thrown upwards at 28 feet per second can be modeled by the function where the height h(x) is given in feet and the time x is given in seconds. At what time in its flight is the ball within 10 feet of the ground? Answer: The ball is within 10 feet of the ground for the first 0.5 second of its flight and again after 1.25 seconds until the ball hits the ground. 11/30/2018 Example 7-4b

First, solve the related equation . Solve algebraically. First, solve the related equation . Related quadratic equation Subtract 2 from each side. Factor. Zero Product Property or Solve each equation. 11/30/2018 Example 7-5a

Plot –2 and 1 on a number line Plot –2 and 1 on a number line. Use closed circles since these solutions are included. Notice that the number line is separated into 3 intervals. 11/30/2018 Example 7-5a

Test a value in each interval to see if it satisfies the original inequality. 11/30/2018 Example 7-5a

Answer: The solution set is This is shown on the number line below. 11/30/2018 Example 7-5a

Solve algebraically. Answer: 11/30/2018 Example 7-5b

Homework - Honors See Syllabus 6.7 pp. 333-334: 14, 24, 26, 28, 32-40 even 43-49 11/30/2018

Homework See Syllabus 6.5 pp. 333-334: 15 – 39 multiples of 3 11/30/2018