Lesson 8.1 Exercises, pages

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1 Lesson 8.1 Eercises, pages 1 9 A. Complete each table of values. a) To complete the table for 3, take the absolute value of each value of 3. b) To complete the table for, take the absolute value of each value of. 5. The graph of = f () is given. Sketch the graph of = ƒ f () ƒ. a) b) f () f () f () f () 5 The value of is never negative, so reflect, in the -ais, the part of each graph that is below the -ais.. Sketch a graph of each absolute value function. a) = ƒ -3 ƒ b) = ƒ ƒ 3 3 and Graph the line 3. The graph lies below the -ais, so to graph 3, reflect the entire graph in the -ais. Graph the line. The graph lies above the -ais, so the graph of is the same as the graph of. 1

2 B 7. Complete each table of values, then graph the absolute value function. Identif its intercepts, domain, and range. a) Plot the points, then join them. From the graph, the -intercept is.5 and the -intercept is. The domain of is ç, and the range is». b) Plot the points, then join them with a smooth curve. From the graph, the -intercepts are and, and the -intercept is 1. The domain of 1 is ç, and the range is». DO NOT COPY. P

3 8. Write each absolute value function in piecewise notation. a) b) 3 9 ( 1) + 1 -intercept: when 3 9», or 3 ( 3 9), or 3 9 when 3 9<, or >3 3 9, if 3 e 3 9, if >3 -intercepts: and The graph of ( 1) 1 opens down. ( 1) 1» for. So, for these domain values, ( 1) 1 ( 1) 1. When < or >, the value of ( 1) 1 is negative. So, for these domain values, ( 1) 1 ( ( 1) 1), or ( 1) 1. e ( 1) 1, if ( 1) 1, if < or > 9. Write each absolute value function in piecewise notation. a) = ƒ - + ƒ b) = ƒ 3-5 ƒ when»» ( ), or when < < >, if e, if > 3 5 when 3 5» 3» 5» 5 3 (3 5), or 3 5 when 3 5< < , if» 5 3 c 3 5, if < 5 3 3

4 c) = ƒ ( - ) ƒ d) = ƒ -( - 1) + ƒ Determine the -intercepts of the graph of ( ). ( ) The -intercepts are: and The graph opens up, so between the -intercepts, the graph is below the -ais. For the graph of ( ) : For or», the value of ( )» For <<, the value of ( )<; that is, ( ). ( ), if or» e ( ), if << Determine the -intercepts of the graph of ( 1). ( 1) ( 1) ( 1) The -intercepts are: 1 and 3 The graph opens down, so between the -intercepts, the graph is above the -ais. For the graph of ( 1) : For 1 3, the value of ( 1)» For < 1 or >3, the value of ( 1) <; that is, ( ( 1) ), or ( 1), e ( 1), if 1 3 ( 1), if < 1 or >3 1. Sketch a graph of each absolute value function. Identif the intercepts, domain, and range. a) = ƒ + 1 ƒ b) = ƒ ƒ Draw the graph of 1. Draw the graph of 3 3. It has -intercept.5 It has -intercept 1. Reflect, in the -ais, the part of Reflect, in the -ais, the part of the graph that is below the -ais. the graph that is below the -ais. From the graph, the -intercept From the graph, the -intercept is 1, is.5, the -intercept is 1, the the -intercept is 3, the domain of domain of 1 is ç, 3 3 is ç, and the and the range is». range is». DO NOT COPY. P

5 c) = ƒ ( + ) + 1 ƒ d) = ƒ - + ƒ ( ) 1 The graph of ( ) 1 opens up with verte (, 1), so the graph has no -intercepts. It has -intercept 5. The graph lies above the -ais, so the graph of ( ) 1 is the same as the graph of ( ) 1. From the graph, there are no -intercepts, the -intercept is 5, the domain of ( ) 1 is ç and the range is» 1. Draw the graph of. The graph opens down and has -intercepts 1 and 1. The verte is at (, ). Reflect, in the -ais, the part of the graph that is below the -ais. From the graph, the -intercepts are 1 and 1, the -intercept is. The domain of is ç and the range is». 11. For each description of a quadratic function = f (), sketch a graph of = ƒ f () ƒ. Describe our thinking. a) the graph of = f () opens up and its verte is: i) on the -ais ii) above the -ais f() f() The graph of f() is The graph of f() is on or above the -ais so above the -ais so the graphs of f() and the graphs of f() and f() are the same. f() are the same. iii) below the -ais Part of the graph of f() is below the -ais so this part of the graph is reflected in the -ais to get the graph of f(). f() 5

6 b) the graph of = f () opens down and its verte is: i) on the -ais ii) above the -ais f() f() The graph of f() is Part of the graph of f() is on or below the -ais so below the -ais so this part the graph of f() is reflected of the graph is reflected in the in the -ais to get the graph -ais to get the graph of f(). of f(). iii) below the -ais The graph of f() is below the -ais so the graph is reflected in the -ais to get the graph of f(). f() 1. The function = f () is linear. Two points on the graph of the absolute value function = ƒ f ()ƒ are (3, ) and (-1, ). Determine the value of = ƒ f ()ƒ when =-3. Eplain our strateg. f() ( 1, ) is the critical point. Plot the points (3, ) and ( 1, ) and draw a line through them. The graph of f() is smmetrical about the line 1. So, reflect the point (3, ) in the line 1, then draw a line through ( 1, ) and the reflected point, ( 5, ), to complete the graph of f(). From the graph, when 3, the value of is 1. DO NOT COPY. P

7 13. This graph represents the volume of water in a 1-L tank over a period of time. Volume (L) 1 Volume of Water in a Tank 8 v 8 Time (h) t a) Write the function represented b this graph in piecewise notation. Let v represent the volume in litres and t represent the time in hours. Equation of line segment for t : v-intercept: 1 t-intercept: Equation has form: v mt 1 Use the points (,1) and (, ) to determine the slope. 1 m 3 So, equation is: v 3t 1 Equation of line segment for <t 8: Passes through: (, ) and (8, 1) Equation has form: v mt b Slope: m Use: v 3t b Substitute: t, v 3() b b 1 So, equation is: v 3t 1 3t 1, if t v e 3t 1, if <t 8 b) Which absolute value function is represented b this graph? How do ou know? The equation of the absolute value function is the absolute value of the left segment or the absolute value of the right segment: v 3t 1 or v 3t 1 7

8 c) Determine the volume of water in the tank after 5.5 h. What strateg did ou use? What other strateg could ou use? Substitute t 5.5 in v 3t 1 : v 3(5.5) 1 v.5 v.5 The volume of water in the tank is.5 L. I could also use the graph to estimate the volume of water after 5.5 h, but this would be an approimate value. C 1. Two linear functions = f () and = h() have opposite slopes and the same -intercept. How will the graphs of = ƒ f () ƒ and = ƒ h() ƒ compare? Include sketches in our eplanation. Both functions are linear so use the slope-point form of a line. f(): m( a) : m( a) g(): m( a) : m( a) Since m( a) m( a), the graphs of f() and h() are the same. For eample, the graphs of and have the same -intercepts, but opposite slopes. To graph the absolute value of each function, I reflect the part of the graph below the -ais in the -ais. So, the graphs of and will be the same. 15. Write an equation for each absolute value function. Eplain our strateg. a) Determine the equation of the line on the right: -intercept: 3 -intercept: 3 Equation has form: m 3 Use the points (, 3) and ( 3, ) to determine the slope. m So, an equation is: 3 and an equation for the absolute value function is: 3 8 DO NOT COPY. P

9 b) 8 Either the middle piece of the graph or the two end pieces were reflected in the -ais. I will assume the middle piece was reflected. So, the graph of the quadratic function opens up and has verte (1, 9). So, the equation has the form a( 1) 9. Use the point (, ) to determine a. Substitute and. a( 1) 9 9 9a a 1 So, an equation of the quadratic function is ( 1) 9 and an equation of the absolute value function is ( 1) 9. 9