Copyright © 2011 Pearson Education, Inc. Slide Absolute Value Functions: Graphs, Equations, Inequalities, and Applications Recall: Use this concept to define the absolute value of a function f : Technology Note: The command abs(x) is used by some graphing calculators to find absolute value...
Copyright © 2011 Pearson Education, Inc. Slide The Graph of y = | f (x)| To graph the function the graph is the same as for values of that are nonnegative and reflected across the x-axis for those that are negative. The domain of is the same as the domain of f, while the range of will be a subset of Example Give the domain and range of Solution
Copyright © 2011 Pearson Education, Inc. Slide Use the graph of to sketch the graph of Give the domain and range of each function. Solution 2.4 Sketch the Graph of y = | f (x)| Given y = f (x)
Copyright © 2011 Pearson Education, Inc. Slide Properties of Absolute Value For all real numbers a and b, Example Consider the following sequence of transformations.
Copyright © 2011 Pearson Education, Inc. Slide Comprehensive Graph We are often interested in absolute value functions of the form where the expression inside the absolute value bars is linear. We will solve equations and inequalities involving such functions. The comprehensive graph of will include all intercepts and the lowest point on the “V-shaped” graph.
Copyright © 2011 Pearson Education, Inc. Slide Equations and Inequalities Involving Absolute Value Example Solve Analytic Solution For to equal 7, 2x + 1 must be 7 units from 0 on the number line. This can only happen when Graphing Calculator Solution
Copyright © 2011 Pearson Education, Inc. Slide Let k be a positive number. Case 1 To solve solve the compound equation Case 2 To solve solve the compound inequality Case 3 To solve solve the three-part inequality Inequalities involving are solved similarly, using the equality part of the symbol as well. 2.4 Solving Absolute Value Equations and Inequalities
Copyright © 2011 Pearson Education, Inc. Slide Solving Absolute Value Inequalities Analytically Solve the inequalities
Copyright © 2011 Pearson Education, Inc. Slide Solving Absolute Value Inequalities Graphically Solve the previous equations graphically by letting and and find all points of intersection. (a)The graph of lies below the graph of for x-values between –4 and 3, supporting the solution set (–4,3). The graph of lies above the graph of for x-values greater than 3 or less than –4, confirming the analytic result.
Copyright © 2011 Pearson Education, Inc. Slide Solving Special Cases of Absolute Value Equations and Inequalities Solve Analytically (a)Because the absolute value of an expression is never negative, the equation has no solution. The solution set is Ø. (b)Using similar reasoning as in part (a), the absolute value of an expression will never be less than –5. The solution set is Ø. (c)Because absolute value will always be greater than or equal to 0, the absolute value of an expression will always be greater than –5. The solution set is Graphical Solution The graphical solution is seen from the graphing of
Copyright © 2011 Pearson Education, Inc. Slide Solving |ax + b| = |cx + d| Analytically Example Solve To solve the equation analytically, solve the compound equation
Copyright © 2011 Pearson Education, Inc. Slide Solving |ax + b| = |cx + d| Graphically Solve Let The equation is equivalent to so graph and find the x-intercepts. From the graph below, we see that they are –1 and 9, supporting the analytic solution.
Copyright © 2011 Pearson Education, Inc. Slide Solving Inequalities Involving Two Absolute Value Expressions Solve each inequality graphically. Solution (a)The inequality In the previous example, note that the graph of is below the x-axis in the interval (b)The inequality is satisfied by the closed interval
Copyright © 2011 Pearson Education, Inc. Slide Solving an Equation Involving a Sum of Absolute Values Solve graphically by the intersection-of-graphs method. Solution Let The points of intersection of the graphs have x-coordinates –9 and 7. To verify these solutions, we substitute them into the equation. Therefore, the solution set is