Chapter 8 Quadratic Functions Copyright 2015 2011 2007

Chapter 8 Quadratic Functions Copyright © 2015, 2011, 2007 Pearson Education, Inc. Chapter 8 -

Chapter Sections 8. 1 – Solving Quadratic Equations by Completing the Square 8. 2 – Solving Quadratic Equations by the Quadratic Formulas 8. 3 – Quadratic Equations: Applications and Problem Solving 8. 4 – Writing Equations in Quadratic Form 8. 5 – Graphing Quadratic Functions 8. 6 – Quadratic and Other Inequalities in One Variable Copyright © 2015, 2011, 2007 Pearson Education, Inc. Chapter 8 -2 2

§ 8. 6 Quadratic and Other Inequalities in One Variable Copyright © 2015, 2011, 2007 Pearson Education, Inc. Chapter 8 -

Quadratic and Other Inequalities Quadratic Inequality A quadratic inequality is an inequality that can be written in one of the following forms ax 2 + bx + c < 0 ax 2 + bx + c > 0 ax 2 + bx + c ≤ 0 ax 2 + bx + c ≥ 0 where a, b, and c are real numbers, with a ≠ 0. Solution to a Quadratic Inequality The solution to a quadratic inequality is the set of all values that make the inequality a true statement. Copyright © 2015, 2011, 2007 Pearson Education, Inc. Chapter 8 -4 4

Solve Quadratic Inequalities To Solve Quadratic and Other Inequalities 1. Write the inequality as an equation and solve the equation. The solutions are the boundary values. 2. Construct a number line and mark each boundary value from step 1 as follows • If the inequality symbol is < or >, use an open circle ◦. • If the inequality symbol is ≤ or ≥, use a closed circle, ●. 3. If solving a rational inequality, determine the values that make any denominator 0. These values are also boundary values. Indicate these boundary values on your number line with an open circle. Copyright © 2015, 2011, 2007 Pearson Education, Inc. Chapter 8 -5 5

Solve Quadratic Inequalities To Solve Quadratic and Other Inequalities 4. Select a test value in each interval and determine whether it satisfies the inequality. 5. The solution is the set of points that satisfy the inequality. 6. Write the solution in the form instructed. Copyright © 2015, 2011, 2007 Pearson Education, Inc. Chapter 8 -6 6

Solve Quadratic Inequalities Example Solve the inequality x 2 – 4 x ≥ -4. Give the solution a) on a number line, b) in interval notation, and c) in set builder notation. continued Copyright © 2015, 2011, 2007 Pearson Education, Inc. Chapter 8 -7 7

Solve Quadratic Inequalities The solution set includes both intervals and the boundary value, 2. The solution set is the set of real numbers. a) b) Copyright © 2015, 2011, 2007 Pearson Education, Inc. c) Chapter 8 -8 8

Solve Other Polynomial Inequalities The same procedure we used to solve quadratic inequalities can be used to solve other polynomial inequalities, as illustrated in the following examples. Example Solve the polynomial inequality (3 x – 2)(x + 3)(x + 5) < 0. Illustrate the solution on a number line and write the solution in both interval notation and set builder notation. Copyright © 2015, 2011, 2007 Pearson Education, Inc. Chapter 8 -9 9

Solve Other Polynomial Inequalities We use the zero-factor property to solve the equation. or or The boundary values -5, -3, and 2/3 are indicated with open circles and break the number line into four intervals. The test values we will use are -6, -4, 0, and 1. We show the results in the following table. continued Copyright © 2015, 2011, 2007 Pearson Education, Inc. Chapter 8 -10 10

Solve Other Polynomial Inequalities The solutions are Copyright © 2015, 2011, 2007 Pearson Education, Inc. Chapter 8 -11 11

Solve Rational Inequalities Rational inequalities are inequalities that contain at least one rational expression. Example Solve the inequality . Graph the solution on a number line and write the solution in interval notation. continued Copyright © 2015, 2011, 2007 Pearson Education, Inc. Chapter 8 -12 12

Solve Rational Inequalities We start by solving the equation . Thus, -7 is a boundary value and is indicated with a closed circle on the number line. continued Copyright © 2015, 2011, 2007 Pearson Education, Inc. Chapter 8 -13 13

Solve Rational Inequalities When solving rational inequalities, we also need to determine the value or values that make the denominator 0. We set the denominator equal to 0 and solve. Therefore, -3 is a boundary value. We use the solution to the equation, -7, and the value that makes the denominator 0, -3, as our boundary values. We will use -8, -5, and 0 as our test values. continued Copyright © 2015, 2011, 2007 Pearson Education, Inc. Chapter 8 -14 14

Solve Rational Inequalities The solution written in interval notation is [-7, -3), and the solution illustrated on the number line is continued Copyright © 2015, 2011, 2007 Pearson Education, Inc. Chapter 8 -15 15

Solve Rational Inequalities The graph below shows of y=2. and the graph Copyright © 2015, 2011, 2007 Pearson Education, Inc. Chapter 8 -16 16