1.3. Absolute Value and Piecewise-Defined Functions Absolutely Piece-ful. My Notes ACTIVITY

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1 Absolute Value and Piecewise-Defined Functions Absolutel Piece-ful SUGGESTED LEARNING STRATEGIES: Activating Prior Knowledge, Create Representations, Quickwrite. Graph both = - for < 3 and = for 3 on the same coordinate grid. M Notes ACTIVITY.3. Describe the graph in Item as completel as possible. Wh is the graph a function? 00 College Board. All rights reserved. 3. Graph = - 3 for 0 and = + for > 0 on the same coordinate grid.. Describe the graph in Item 3 as completel as possible. Wh is the graph a function? Unit Linear Sstems and Matrices 3

2 ACTIVITY.3 Absolute Value and Piecewise-Defined Functions Absolutel Piece-ful M Notes ACADEMIC VOCABULARY A piecewise-defined function is a function that is defined using different rules for the different non-overlapping intervals of its domain. SUGGESTED LEARNING STRATEGIES: Marking the Tet, Interactive Word Wall, Create Representations, Activating Prior Knowledge The functions in Items and 3 are piecewise-defined functions. Piecewise-defined functions are written as follows (using the function from Item 3 as an eample): Name of function f () = { - 3 Single Brace + if 0 if > 0 Domain Restrictions Function Rules 5. Complete the table of values. Then graph the function. g () = { if < - if - A piecewise-defined function ma have more than two rules. For eample, consider the function below. h( ) = - if < - if - < if g () The domain of a piecewise-defined function consists of the union of all the domains of the individual pieces of the function. Likewise, the range of a piecewise-defined function consists of the union of all the ranges of the individual pieces of the function.. What is the domain and range of g () in Item 5? 00 College Board. All rights reserved Another tpe of piecewise-defined function is a step function. One such function is the greatest integer function, written f() = [], which ields a value f() that is the greatest integer less than or equal to the value of. For eample, f(.7) = [.7] = because the greatest integer less than or equal to.7 is ; and f(-3.) = [-3.] = - because the greatest integer less than or equal to -3. is -. One step of this function is graphed to the left. Complete the graph. SpringBoard Mathematics with Meaning Algebra

3 Absolute Value and Piecewise-Defined Functions Absolutel Piece-ful ACTIVITY.3 SUGGESTED LEARNING STRATEGIES: Create Representations, Quickwrite, Self/Peer Revision, Marking the Tet, Interactive Word Wall TRY THESE A Graph each function and identif its domain. Show our work. a. f () = { + - if < 0 if 0 b. g () = { if < if > M Notes 7. Complete the table and graph the piecewise-defined function. Identif the domain and range of the function. f () = { - if < 0 if 0 00 College Board. All rights reserved f () 8. Describe the graph of f () shown in Item 7. Identif as man characteristics as possible. ACADEMIC VOCABULARY An absolute value function is written as f() = and is defined b f () = { - if < 0 if 0 The function f () in item 7 is known as the absolute value function. The notation for the function is f () =. The sharp change in the graph at = 0 is the verte. CONNECT TO AP The verte of an absolute value function is an eample of a cusp in a graph. A graph has a cusp at a point where there is an abrupt change in direction. Unit Linear Sstems and Matrices 5

4 ACTIVITY.3 Absolute Value and Piecewise-Defined Functions Absolutel Piece-ful M Notes SUGGESTED LEARNING STRATEGIES: Marking the Tet, Interactive Word Wall, Activating Prior Knowledge, Create Representations 9. Use the piecewise definition of the absolute value function to evaluate each epression. a. f (-) = b. f (8) = c. f (0) = d. f ( - 5 ) = 0. Could ou have determined the values of the function in Item 9 another wa? Eplain our reasoning. ACADEMIC VOCABULARY parent function ACADEMIC VOCABULARY A transformation is a change in the position, size, or shape of a parent graph. MATH TERMS Transformations include: vertical translations, which shift a graph up or down vertical stretches or vertical shrinks, which stretch or shrink a graph reflections, which produce a mirror image of a graph over a line The absolute value function f () = is the parent absolute value function. Recall that a parent function is the most basic function of a particular tpe. Transformations ma be performed on a parent function to produce a new function.. For each function below, graph the function and identif the transformation(s) of f () =. a. g () = + b. h () = - 00 College Board. All rights reserved. SpringBoard Mathematics with Meaning Algebra

5 Absolute Value and Piecewise-Defined Functions Absolutel Piece-ful ACTIVITY.3 SUGGESTED LEARNING STRATEGIES: Activating Prior Knowledge, Create Representations, Predict and Confirm. () M Notes c. k () = 3 d. q () = - + TECHNOLOGY Man graphing calculators use a function called abs to represent absolute value.. Use the coordinate grid at the right. a. Graph the parent function f () =. b. Predict the transformation for g () = College Board. All rights reserved. c. Graph the function g () = - 3. d. What transformation does our graph show? 3. Use the results from Item to predict the transformation of h() = +. Then graph the function to confirm or revise our prediction. Unit Linear Sstems and Matrices 7

6 ACTIVITY.3 Absolute Value and Piecewise-Defined Functions Absolutel Piece-ful M Notes MATH TERMS A horizontal translation is a transformation that moves the graph of a function to the right or to the left without changing the shape of the original graph. SUGGESTED LEARNING STRATEGIES: Create Representations, Predict and Confirm, Marking the Tet, Interactive Word Wall, Look for Pattern The functions in Items and 3 are eamples of horizontal translations. A horizontal translation occurs when the independent variable,, is replaced with + a or with - a.. In the absolute value function f () = + a with a > 0, describe how the graph of the function changes, compared to the parent function. 5. In the absolute value function f () = - a with a > 0, describe how the graph of the function changes, compared to the parent function. TRY THESE B Graph each function. a. f () = - b. f () = For each absolute value function, describe the transformations represented in the rule and use them to graph the function. a. g () = Description of transformations 00 College Board. All rights reserved. 8 SpringBoard Mathematics with Meaning Algebra

7 Absolute Value and Piecewise-Defined Functions Absolutel Piece-ful ACTIVITY.3 SUGGESTED LEARNING STRATEGIES: Predict and Confirm, Create Representations M Notes b. h() = Description of transformations c. k() = Description of transformations 00 College Board. All rights reserved. 7. The graph at the right shows = and =. You can write an equation in one variable using the two given equations. How do the points of intersection of the two graphs relate to the equation =? = 5 = Unit Linear Sstems and Matrices 9

8 ACTIVITY.3 Absolute Value and Piecewise-Defined Functions Absolutel Piece-ful M Notes Recall that the geometric interpretation of is the distance from the number to 0 on a number line. If = 5, then = -5 or = 5 because those two values are both 5 units awa from 0 on a number line. SUGGESTED LEARNING STRATEGIES: Quickwrite, Self/Peer Revision, Marking the Tet, Interactive Word Wall You can use the definition of absolute value to solve absolute value equations algebraicall. -(a + b), a + b < 0 Since a + b = { a + b, a + b 0, then the equation a + b = c is equivalent to -(a + b) = c or (a + b) = c. Since -(a + b) = c is equivalent to a + b = -c, then the absolute value equation a + b = c is equivalent to a + b = -c or a + b = c. EXAMPLE Solve =. Step : Isolate the absolute value epression. Add 5 to both sides and then divide b. Step : Write and solve two equations using the definition of absolute value = - = - = 3 - = 3 or - = -3 = or = - Solution: There are two solutions: = and = - Check to see if both solutions satisf the original equation. Substitute and - for in the original equation = 3-5 = (3) - 5 = - 5 = TRY THESE C = -3-5 = (3) - 5 = - 5 = Solve each absolute value equation. Show our work in the M Notes space. a. - = 3 b. + - = - c = d = 00 College Board. All rights reserved. 30 SpringBoard Mathematics with Meaning Algebra

9 Absolute Value and Piecewise-Defined Functions Absolutel Piece-ful ACTIVITY.3 SUGGESTED LEARNING STRATEGIES: Identif a Subtask, Marking the Tet, Interactive Word Wall 8. How man solutions are possible for an absolute value equation? M Notes Solving absolute value inequalities algebraicall is similar to solving absolute value equations. Using the definition of absolute value, ou know that a + b > c, where c > 0, is equivalent to -(a + b) > c or (a + b) > c. Multipling the first inequalit b -, and then using a similar method for a + b < c, gives these statements. a + b > c, c > 0, is equivalent to a + b < -c or a + b > c. a + b < c, c > 0, is equivalent to a + b < c or a + b > -c, which can also be written as -c < a + b < c. EXAMPLE Solve >. Graph the solutions on a number line. Step : Isolate the absolute value epression > + 3 > 5 Step : Write two inequalities + 3 > 5 or + 3 < -5 Step 3: Solve each inequalit. > or < - Solution: 0 00 College Board. All rights reserved. EXAMPLE 3 Solve < 7. Graph the solutions on a number line. Step : Isolate the absolute value epression < < Step : Write the compound inequalit - < 3 - < Step 3: Solve the inequalit. - 3 < < Solution: TRY THESE D Solve and graph each absolute value inequalit. 3 a. - > 3 b c d < Unit Linear Sstems and Matrices 3

10 ACTIVITY.3 Absolute Value and Piecewise-Defined Functions Absolutel Piece-ful M Notes SUGGESTED LEARNING STRATEGIES: Identif a Subtask, Quickwrite, Self/Peer Revision 9. Wh is the condition c > 0 necessar for a + b < c to have a solution? CHECK YOUR UNDERSTANDING Write our answers on notebook paper or grid paper. Show our work.. Graph each piecewise-defined function. Then identif the domain and range. a. f () = { if 0 if > 0 b. f () = 3 if < - { - + if -. A step function known as the ceiling function ields the value g() that is the least integer greater than or equal to. Graph this step function. 3. Graph each transformation of f () =. a. f () = - + b. f () = - 3 c. f () = d. f () = Solve each absolute value equation. a. - = 5 b. 3-7 = c = 0-5 d = 5 e = 5. Write the equation for each transformation of f () = described below. a. stretch verticall b a factor of 5, translate left 9 units, and translate down 3 units. b. reflect over the -ais, stretch verticall b a factor of, and translate left units. c.. Solve each absolute value inequalit. Graph the solutions on a number line. a. - 7 > b c d < 5 e MATHEMATICAL REFLECTION How does the definition of absolute value as a piecewise-defined function relate to the method of solving absolute value equations? 5 00 College Board. All rights reserved. 3 SpringBoard Mathematics with Meaning Algebra