OBJECTIVES: F.IF.B.4 Interpret functions that arise in applications in terms of the context. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. F.IF.B.7 Analyze functions using different representations. Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases. b. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. BIG IDEA: Absolute-value graphs are V-shaped graphs made up of two linear pieces that are symmetrical across an axis of symmetry. The axis of symmetry is a vertical line that runs through the. All absolute value function graphs are translations of the parent function s graph; f ( x) x. PREREQUISITE SKILLS: students should understand how to evaluate absolute-value expressions students should understand how to solve and graph equations in two variables students should understand how to read and write equations in function notation VOCABULARY: absolute-value function: a function whose rule contains an absolute-value expression axis of symmetry: a line that divides the graph into two symmetrical halves : the corner point on a graph SKILLS: evaluate and graph absolute value equations find possible solutions to those equations using coordinate geometry Alg Unit 04c Notes Absolute Value Functions Page of 6/0/03
REVIEW AND EXAMPLES: Absolute Value Function: a function that can be written in the form f ( x) a x h k Ex. Use a table of values to graph the function f ( x) f ( x) x 4 0 4 f(x) 4 0 4 x. x is the parent function. Notice that the parent graph is V-shaped, opens up and the of graph is on the origin. Ex. Use a table of values to graph the function g( x) x 4 0 4 g(x) 4 0 4 x and the parent function is that a = - (the opposite of the parent function). The difference between this graph and the parent graph is that it has been reflected across the x-axis, so it opens down. Note: It is important to make sure to find the of an absolute-value equation to avoid graphing one side of the function only. To do this, make sure to start the table of values in the middle with the x value that gives 0 for the expression inside the absolute value symbols. Then assign x values on each side of that first value. Ex 3. Use a table of values to graph the function g( x) x 3. x 4 0 4 g(x) 7 5 3 5 7 and the parent function is that 3 is being added to the absolute value expression. The difference between this graph and the parent graph is that it has been translated up 3 units. Alg Unit 04c Notes Absolute Value Functions Page of 6/0/03
Ex 4. Use a table of values to graph the function g( x) x. x 4 0 4 g(x) 3 3 and the parent function is that is being subtracted from the absolute value expression. The difference between this graph and the parent graph is that it has been translated down unit. Ex 5. Use a table of values to graph the function g( x) x 4. x 8 6 4 0 g(x) 4 0 4 and the parent function is that 4 is being added to the variable inside of the absolute value expression. The difference between this graph and the parent graph is that it has been translated left 4 units. Ex 6. Use a table of values to graph the function g( x) x x 0 4 6 g(x) 4 0 4 and the parent function is that is being subtracted from the variable inside of the absolute value expression. The difference between this graph and the parent graph is that it has been translated right units. Alg Unit 04c Notes Absolute Value Functions Page 3 of 6/0/03
Ex 7. Use a table of values to graph the function g( x) x x 4 0 4 g(x) 8 4 0 4 8 and the parent function is that a = (the a value has doubled). The difference between this graph and the parent graph is that it has been stretched (narrower); the rate of change from each domain value to the next has doubled. * Since a =, he slope is to the left of the and to the right of the. Ex 8. Use a table of values to graph the function x 4 0 4 g(x) 0.5 0 0.5 g( x) 4 x and the parent function is that a = 4 (the a value is one quarter the size). The difference between this graph and the parent graph is that it has been shrunk (wider); the rate of change from each domain value to the next is one quarter the amount. * Since a = 4, the slope is to the left of the and to the right of the. 4 4 Alg Unit 04c Notes Absolute Value Functions Page 4 of 6/0/03
Ex 9. Use a table of values to graph the function g( x) 4 x x 0 g(x) 8 4 0 4 8 and the parent function is that a = 4 (the a value is the opposite of four times the size). The difference between this graph and the parent graph is that it has been reflected across the x-axis and it has stretched (narrower); the rate of change from each domain value to the next is the opposite of four times the amount. * Since a = 4, the slope is 4 to the left of the and 4 to the right of the. Ex 0. Use a table of values to graph the function x 4 0 4 g(x) 0 g( x) x and the parent function is that a = (the a value is the opposite of half of the size). The difference between this graph and the parent graph is that it has been reflected across the x-axis and it has shrunk (wider); the rate of change from each domain value to the next is the opposite of half of the amount. * Since a =, the slope is to the left of the and to the right of the. Alg Unit 04c Notes Absolute Value Functions Page 5 of 6/0/03
Conclusions: Describe the effects of a, h, and k in the equation f ( x) a x h k on the graph of f ( x) x the Parent Function. a: if, a 0 then, the graph opens-down V (reflected across x-axis) if, a then, the graph shrinks, it s wider then f ( x) if, a then, the graph stretches, it s narrower then f ( x) h: if, h 0 then, the graph shifts h units to the left (x-coordinate of = h) if, h 0 then, the graph shifts h units to the right (x-coordinate of = h) k: if, k 0 then, the graph shifts k units down (y-coordinate of = k) if, k 0 then, the graph shifts k units up (y-coordinate of = k) Note: (h, k) is always the of the graph of the function. Ex. Without graphing, find the of the absolute value function f ( x) x 5. h 5, so the x-coordinate of the is 5. k, so the y-coordinate of the is. The is 5,. Ex. Graph the absolute value function Transformations on f ( x) Find the : h 0 and k 3 f ( x) x 3 x : reflect over x-axis, (opens down) wider, since a graph shifted up 3, so the is 0,3 x x *Since a =, the slope is to the left of the and to the right of the. Ex 3. Graph the absolute value function y x 3 5. Transformations on f ( x) Find the : h 3 and k 5 x : narrower, since a graph shifted left 3 graph shifted down 5, so the is 3, 5 * Since a =, the slope is to the left of the and to the right of the. Alg Unit 04c Notes Absolute Value Functions Page 6 of 6/0/03
Solving Absolute Value Equations Graphically When we solve any function for x, what we are really solving for are the x-intercepts of the function s graph (roots). Ex 4. Solve the equation 3 x 9 graphically and algebraically. Graphically Step : Set the equation equal to zero and write the related function rule. 3 x 9 9 9 3 x 9 0 f ( x) 3 x 9 Step : Graph the absolute value function. is at (, 9) a = 3,so slope is 3 to the left of the and 3 to the right of the Step 3: Find the x-intercepts (roots) of the graph. Solutions: x =, 5 Algebraically: Step : Isolate the absolute value expression 3 x 9 3 3 x 3 Step : Solve the absolute value equation. x = 3 and x = 3 x = 5 and x = Alg Unit 04c Notes Absolute Value Functions Page 7 of 6/0/03
ASSESSMENT ITEMS:. Graph the absolute value function f x x 4. 3 ANS: Step : graph the. h = 4 and k =, so the is (4, ) Step : graph the V-shaped graph a =,so the slope is to the left of the 3 3 and to the right of the 3. Describe graphically why an absolute value equation can have two, one, or no solutions. ANS: It depends on whether the graph opens up or down and where the lies with respect to the x-axis. *There are two ways that the function would have two solutions: If the graph opens up and the lies below the x-axis, then the graph will cross the x-axis in two places and the function would have two solutions. Similarly, if the graph opens down and the lies above the x-axis, the graph will cross the x-axis in two places and the function would have two solutions. *There is one way that the function would have exactly one solution: If the falls on the x-axis the function would have one solution. *There are two ways that the function would have no solutions: If the graph opens up and the lies above the x-axis, then the graph will never cross the x-axis and the function would have no solutions. Similarly, if the graph opens down and the lies below the x- axis, the graph will never cross the x-axis and the function would have no solutions. 3. Determine if the function f ( x) 3 x 5 opens up or down and find the. A. opens up; is at (, 5) B. opens up; is at (, 5) C. opens down; is at (, 5) D. opens down; is at (, 5) ANS: D Alg Unit 04c Notes Absolute Value Functions Page 8 of 6/0/03
4. Use the graph below. What is the equation of the function? A. y x B. y x C. y x D. y x ANS: B 5. If an absolute value function s graph is narrower than that of the parent graph, what do you know about the function? A. the graph opens down B. h is negative C. a D. a 6. State the domain and the range of the function, given its graph. ANS: domain: all real numbers range: y 3 The is the maximum point on the graph. ANS: C Alg Unit 04c Notes Absolute Value Functions Page 9 of 6/0/03
7. Which function below has a graph that has been translated right from the parent function s graph? A. f ( x) x 4 B. g( x) x 3 C. h( x) x 6 D. k( x) x 5 ANS: D 8. Which function below has a graph that has been narrowed and reflected across the x-axis when compared to the parent function s graph? A. f ( x) x 4 B. g( x) x 3 C. h( x) x 6 D. k( x) x 5 9. What is the axis of symmetry of the graph of f ( x) 4 x? A. x = 4 B. x = 0 C. x = D. x = ANS: A ANS: B 0. Is the of the graph of graph. Justify your answer. h( x) x 4 the minimum or maximum point on the 3 ANS: It s the minimum point of the graph. The is at (, 4), which is below the x-axis and since a is positive we know that graph opens up.. If you know the of an absolute value function is (, 4) and a > 0, state the range of the graph of the function. ANS: Range: y 4 Alg Unit 04c Notes Absolute Value Functions Page 0 of 6/0/03
. Use the graph below. What is the equation of the function? A. f ( x) x 4 B. f ( x) x 4 C. f ( x) x 4 D. f ( x) x 4 ANS: A 3. Graph the absolute value function f x x 3 4. ANS: Step : graph the. h = 3 and k = 4, so the is ( 3, 4) Step : graph the V-shaped graph a =, so the slope is to the left of the and to the right of the Alg Unit 04c Notes Absolute Value Functions Page of 6/0/03
4. Use the graph below to solve the absolute value function g( x) 3 x 6. ANS: x =, 3 5. For the absolute value function h( x) x, identify the function s domain and range; the of the function s graph; the axis of symmetry of the function s graph; and the function s minimum or maximum value. ANS: domain: all real numbers range: y : (0, ) axis of symmetry: x = 0 maximum value: Alg Unit 04c Notes Absolute Value Functions Page of 6/0/03