Chapter 2: Analysis of Graphs of Functions 2.1 Graphs of Basic Functions and Relations; Symmetry 2.2 Vertical and Horizontal Shifts of Graphs 2.3 Stretching, Shrinking, and Reflecting Graphs 2.4 Absolute Value Functions 2.5 Piecewise-Defined Functions 2.6 Operations and Composition

2. 4. Absolute Value Functions: Graphs, 2.4 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. . .

2.4 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

2.4 Sketch the Graph of y = | f (x)| Given y = f (x) Use the graph of to sketch the graph of Give the domain and range of each function. Solution

2.4 Properties of Absolute Value For all real numbers a and b, Example Consider the following sequence of transformations.

2.4 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.

2.4 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

2.4 Solving Absolute Value Equations and Inequalities 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 Inequalities Analytically Solve the inequalities

2.4 Solving Absolute Value Inequalities Graphically Solve the previous equations graphically by letting and and find all points of intersection. 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.

2.4 Solving Special Cases of Absolute Value Equations and Inequalities Solve Analytically Because the absolute value of an expression is never negative, the equation has no solution. The solution set is Ø. Using similar reasoning as in part (a), the absolute value of an expression will never be less than –5. The solution set is Ø. 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

2.4 Solving |ax + b| = |cx + d| Analytically Example Solve To solve the equation analytically, solve the compound equation

2.4 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.

2.4 Solving Inequalities Involving Two Absolute Value Expressions Solve each inequality graphically. Solution 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

2.4 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