4.9 Graph and Solve Quadratic

Transcription

1 4.9 Graph and Solve Quadratic Inequalities Goal p Graph and solve quadratic inequalities. Your Notes VOCABULARY Quadratic inequalit in two variables Quadratic inequalit in one variable GRAPHING A QUADRATIC INEQUALITY IN TWO VARIABLES To graph a quadratic inequalit, follow these steps: Step Graph the parabola with equation 5 a 2 b c. Make the parabola for inequalities with < or > and for inequalities with or. Step 2 Test a point (, ) the parabola to determine whether the point is a solution of the inequalit. Step 3 Shade the region the parabola if the point from Step 2 is a solution. Shade the region the parabola if it is not a solution. 6 Lesson 4.9 Algebra 2 Notetaking Guide Copright Holt McDougal. All rights reserved.

2 4.9 Graph and Solve Quadratic Inequalities Goal p Graph and solve quadratic inequalities. Your Notes VOCABULARY Quadratic inequalit in two variables An inequalit that can be written in the form < a 2 b c, a 2 b c, > a 2 b c, or a 2 b c Quadratic inequalit in one variable An inequalit that can be written in the form a 2 b c < 0, a 2 b c 0, a 2 b c > 0, or a 2 b c 0 GRAPHING A QUADRATIC INEQUALITY IN TWO VARIABLES To graph a quadratic inequalit, follow these steps: Step Graph the parabola with equation 5 a 2 b c. Make the parabola dashed for inequalities with < or > and solid for inequalities with or. Step 2 Test a point (, ) inside the parabola to determine whether the point is a solution of the inequalit. Step 3 Shade the region inside the parabola if the point from Step 2 is a solution. Shade the region outside the parabola if it is not a solution. 6 Lesson 4.9 Algebra 2 Notetaking Guide Copright Holt McDougal. All rights reserved.

3 Eample Graph a quadratic inequalit Graph Graph The inequalit smbol is, so make the parabola. 2. Test the point (0, 0). So, (0, 0). 3. Shade the region the parabola. Checkpoint Graph > Eample 2 Graph the sstem of quadratic inequalities. > Inequalit Inequalit 2 Solution Graph a sstem of quadratic inequalities. Graph > The graph is the region (but not 2 including) the parabola Graph The graph is the region and including the parabola Identif the region where the two graphs overlap. This region is the graph of the sstem. Copright Holt McDougal. All rights reserved. Lesson 4.9 Algebra 2 Notetaking Guide 7

4 Eample Graph Graph The inequalit smbol is, so make the parabola solid. 2. Test the point (0, 0). 0 2(0) 2 2(0) Graph a quadratic inequalit So, (0, 0) is a solution. 3. Shade the region inside the parabola. (0, 0) Checkpoint Graph > Eample 2 Graph the sstem of quadratic inequalities. > Inequalit Inequalit 2 Solution Graph a sstem of quadratic inequalities Graph > The graph is 2 the region inside (but not 2 including) the parabola $ Graph The graph is the region inside and including the parabola Identif the region where the two graphs overlap. This region is the graph of the sstem. Copright Holt McDougal. All rights reserved. Lesson 4.9 Algebra 2 Notetaking Guide 7

5 Checkpoint Graph the sstem. 2. < Eample 3 Solve Rewrite the inequalit as Then make a table of values Solve a quadratic inequalit using a table Notice that when the values of are between and, inclusive. The solution of the inequalit is. Checkpoint Complete the following eercise. 3. Solve the quadratic inequalit using a table. 8 Lesson 4.9 Algebra 2 Notetaking Guide Copright Holt McDougal. All rights reserved.

6 Checkpoint Graph the sstem. 2. < $ , Eample 3 Solve a quadratic inequalit using a table Solve Rewrite the inequalit as Then make a table of values Notice that when the values of are between 24 and, inclusive. The solution of the inequalit is 24. Checkpoint Complete the following eercise. 3. Solve the quadratic inequalit using a table Lesson 4.9 Algebra 2 Notetaking Guide Copright Holt McDougal. All rights reserved.

7 Eample 4 Solve a quadratic inequalit b graphing Solve The solution consists of the -values for which the graph of lies the -ais. Find the graph s -intercepts b letting 5 0 and using to solve for ø or ø Sketch a parabola that opens and has and as -intercepts. The graph lies the -ais to the left of (and including) 5 and to the right of (and including) 5. The solution is approimatel. Checkpoint Complete the following eercise. 4. Solve the quadratic inequalit b graphing. 2 Copright Holt McDougal. All rights reserved. Lesson 4.9 Algebra 2 Notetaking Guide 9

8 Eample 4 Solve a quadratic inequalit b graphing Solve The solution consists of the -values for which the graph of lies on or below the -ais. Find the graph s -intercepts b letting 5 0 and using the quadratic formula to solve for Ï}} (25) 2 2 4(23)(3) 2(23) Ï} 6 } 26 ø 0.47 or ø Sketch a parabola that opens down and has 0.47 and 22.4 as -intercepts. The graph lies on or below the -ais to the left of (and including) and to the right of (and including) The solution is approimatel 22.4 or Checkpoint Complete the following eercise. 4. Solve the quadratic inequalit b graphing or Copright Holt McDougal. All rights reserved. Lesson 4.9 Algebra 2 Notetaking Guide 9

9 Eample 5 Solve a quadratic inequalit algebraicall Solve 2 2. First, write and solve the equation obtained b replacing with. Write corresponding equation. Write in standard form. Factor. Zero product propert. The numbers are the critical -values of the inequalit 2 2. Plot on a number line, using dots. The critical -values partition the number line into three intervals. Test an -value in each interval to see if it satisfies the inequalit. Test 5 : Test 5 : Test 5 : The solution is. Checkpoint Solve the inequalit algebraicall Homework 20 Lesson 4.9 Algebra 2 Notetaking Guide Copright Holt McDougal. All rights reserved.

10 Eample 5 Solve 2 2. First, write and solve the equation obtained b replacing with Solve a quadratic inequalit algebraicall ( 4)( 2 3) 5 0 Factor. Write corresponding equation. Write in standard form or 5 3 Zero product propert. The numbers 24 and 3 are the critical -values of the inequalit 2 2. Plot 24 and 3 on a number line, using solid dots. The critical -values partition the number line into three intervals. Test an -value in each interval to see if it satisfies the inequalit Test 5 26 : Test 5 0 : (26) 2 (26) (0) 2 (0) Test 5 5 : (5) 2 (5) The solution is 24 or 3. Checkpoint Solve the inequalit algebraicall or 4 Homework 20 Lesson 4.9 Algebra 2 Notetaking Guide Copright Holt McDougal. All rights reserved.