3-2 Solving Linear Equations by Graphing. Solve each equation by graphing. 2x + 6 = 0. f (x. The graph intersects the x-axis at 3. So the solution is

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1 - Solving Linear Equations by Graphing Solve each equation by graphing. + 6 = The related function is f () = f ( ) = f ( ) = f ( ) = f () = f () = f () = f () = + 6 ( ) + 6 ( ) + 6 () + 6 () + 6 () + 6 () + 6 f ( 6 (, f( ) ) (, ) (, ) (, 6) (, ) (, ) (, ) The graph intersects the - The related function is f () = f ( ) = f ( ) = ( ) f ( ) = ( ) f ( ) = ( ) f () = () f () = () f () = () f ( 5 7 (, f( ) ) (, ) (, ) (, ) (, ) (, 5) (, 7) The graph intersects the -ais at. So the solution is Page

2 - Solving Linear Equations The graph intersects the - by Graphing The related function is f () = f ( ) = f ( ) = ( ) f ( ) = ( ) f ( ) = ( ) f () = () f () = () f () = () f ( 5 7 (, f( ) ) (, ) (, ) (, ) (, ) (, 5) (, 7) The graph intersects the -ais at. So the solution is = The related function is f () = f ( ) = f ( ) = ( ) f ( ) = ( ) f () = () f ( 8 f () = () f () = () f () = () 6 (, f( ) ) (, 8) (, ) (, ) (, 6) (, ) Page

3 - Solving Linear Equations by Graphing The graph intersects the -ais at. So the solution is = The related function is f () = f ( ) = f ( ) = ( ) f ( ) = ( ) f () = () f ( 8 f () = () f () = () f () = () 6 (, f( ) ) (, 8) (, ) (, ) (, 6) (, ) The graph intersects the -ais at. So the solution is Solve algebraically 9 + = The related function is f () = 9 f ( ) = + 6 f ( ) = 9( ) + f ( ) = 9( ) + f ( (, f( ) ) (, ) 5 (, 5) f () = 9() + (, ) Page

4 - Solving Linear Equations by Graphing 9 + = The related function is f () = 9 f ( ) = + 6 f ( ) = 9( ) + f ( ) = 9( ) + f ( (, f( ) ) (, ) 5 (, 5) f () = 9() + f () = 9() + f () = 9() + 9 The graph intersects the -ais at (, ) (, ) (, 9). So the solution is Solve algebraically 5 = + 8 The related function is f () = Page

5 - Solving Linear Equations by Graphing 5 = + 8 The related function is f () = The graph does not intersect the -ais. Therefore, this equation has no solution. + = Graph the related function, which is f ( Page 5

6 - Solving Linear Equations by Graphing The graph does not intersect the -ais. Therefore, this equation has no solution. + = Graph the related function, which is f ( The graph does not intersect the -ais. Therefore, this equation has no solution. 5 = Graph the related function, which is f ( Page 6

7 The graph doesequations not intersect -ais. Therefore, this equation has no solution. - Solving Linear bythe Graphing 5 = Graph the related function, which is f ( The graph does not intersect the -ais. Therefore, this equation has no solution. 6 + = Graph the related function, which is f () = Page 7

8 - Solving Linear Equations by Graphing The graph does not intersect the -ais. Therefore, this equation has no solution. 6 + = Graph the related function, which is f () = The graph does not intersect the -ais. Therefore, this equation has no solution. NEWSPAPERS The function represents the weight w in pounds of the papers in Tyrone s newspaper delivery bag after he delivers n newspapers. Find the zero and eplain what it means in the contet of this situation. The related function is f ( (, ).5 (,.5) 5 (, 5) 7.5 (, 7.5) (, ) (, f( ) ) Page 8

9 7.5 (, 7.5) (, ) - Solving Linear Equations by Graphing The graph intersects the pounds. Tyrone must deliver newspapers for his bag to weigh Solve algebraically To find the zero of the function, substitute zero in for w. This means that Tyrone must deliver newspapers for his bag to weigh pounds. Solve each equation by graphing. = 5 The related function is f () = f ( ) = 5 f ( (, f( ) ) f ( ) = ( ) 5 9 (, 9) f ( ) = ( ) 5 7 (, 7) f () = () 5 5 (, 5) f () = () 5 (, ) f () = () 5 (, ) esolutions Powered 5 Manual f (5) =- (5) 5 by Cognero (5, ) Page 9

10 - Solving Linear Equations by Graphing This means that Tyrone must deliver newspapers for his bag to weigh pounds. Solve each equation by graphing. = 5 The related function is f () = f ( ) = 5 f ( f ( ) = ( ) 5 9 f ( ) = ( ) 5 7 f () = () 5 5 f () = () 5 f () = () 5 5 f (5) = (5) 5 (, f( ) ) (, 9) (, 7) (, 5) (, ) (, ) (5, ) The graph intersects the -ais at 5. So the solution is 5 Solve algebraically =+ The related function is f () = f ( ) = + f ( ) = ( ) + f ( ) = () + f ( ) = ( ) + f () = () + f () = () + f () = () + f ( 5 7 (, f( ) ) (, ) (, ) (, ) (, ) (, 5) (, 7) Page

11 - Solving Linear Equations by Graphing =+ The related function is f () = f ( ) = + f ( ) = ( ) + f ( ) = () + f ( ) = ( ) + f () = () + f () = () + f () = () + f ( 5 7 (, f( ) ) (, ) (, ) (, ) (, ) (, 5) (, 7) The graph intersects the -ais at. So the solution is Solve algebraically 5 8 = 6 8 Graph the related function, which is f () = Page

12 - Solving Linear Equations by Graphing 5 8 = 6 8 Graph the related function, which is f () = The graph does not intersect the -ais. Therefore, this equation has no solution. = + Graph the related function, which is f () = Page

13 - Solving Linear Equations by Graphing The graph does not intersect the -ais. Therefore, this equation has no solution. = + Graph the related function, which is f () = The graph does not intersect the -ais. Therefore, this equation has no solution. 6 = The related function is f () = f ( ) = 6 f ( ) = ( ) 6 f () = () 6 f ( 8 6 (, f( ) ) (, 8) (, 6) f () = () 6 (, ) 6 f (6) = (6) 6 (6, ) (9, ) (, ) 9 f (9) = (9) 6 f () = () 6 Page

14 - Solving Linear Equations by Graphing The graph does not intersect the -ais. Therefore, this equation has no solution. 6 = The related function is f () = f ( ) = 6 f ( ) = ( ) 6 f () = () 6 f ( 8 6 (, f( ) ) (, 8) (, 6) f () = () 6 (, ) 6 f (6) = (6) 6 (6, ) (9, ) (, ) 9 f (9) = (9) 6 f () = () 6 The graph intersects the - Solve Algebraically = 7 + The related function is f () = 7 f ( ) = 7 + f ( ) = 7( ) + f ( ) = 7( ) + f ( 8 (, f( ) ) (, 8) (, ) f () = 7() + f () = 7() + f () = 7() + (, ) (, ) (, ) Page

15 - Solving Linear Equations by Graphing = 7 + The related function is f () = 7 f ( ) = 7 + f ( ) = 7( ) + f ( ) = 7( ) + f ( 8 (, f( ) ) (, 8) (, ) f () = 7() + f () = 7() + f () = 7() + The graph intersects the -ais at (, ) (, ) (, ). So the solution is Solve algebraically + = The related function is f () = f ( ) = + f ( f ( ) = ( ) + f ( ) = ( ) + f ( ) = ( ) + 8 f ( 8) = ( 8) f ( 6) = ( 6) + (, f( ) ) (, ) (, ) (, ) ( 8, 6) ( 6, ) Page 5

16 - Solving Linear Equations by Graphing + = The related function is f () = 8 6 f ( ) = + f ( ) = ( ) + f ( ) = ( ) + f ( ) = ( ) + f ( 8) = ( 8) + f ( 6) = ( 6) + f ( ) = ( ) + f () = () + f ( 6 (, f( ) ) (, ) (, ) (, ) ( 8, 6) ( 6, ) (, ) (, ) The graph intersects the -ais at. So the solution is Solve Algebraically 5 5 = 5 + Graph the related function, which is f () = 7 Page 6

17 - Solving Linear Equations by Graphing 5 5 = 5 + Graph the related function, which is f () = 7 The graph does not intersect the -ais. Therefore, this equation has no solution = 7 Graph the related function, which is f ( The graph does not intersect the -ais. Therefore, this equation has no solution. Page 7

18 The graph doesequations not intersect -ais. Therefore, this equation has no solution. - Solving Linear bythe Graphing = 7 Graph the related function, which is f ( The graph does not intersect the -ais. Therefore, this equation has no solution. 8 = Graph the related function, which is f () = The graph does not intersect the -ais. Therefore, this equation has no solution. Page 8

19 The graph doesequations not intersect -ais. Therefore, this equation has no solution. - Solving Linear bythe Graphing 8 = Graph the related function, which is f () = The graph does not intersect the -ais. Therefore, this equation has no solution. = 6 8 The related function is f () = 6 f ( ) = 6 8 f ( ) = 6( ) 8 f ( ) = 6( ) 8 f () = 6() 8 f ( 8 (, f( ) ) (, ) (, ) (, 6) f () = 6() 8 f () = 6() 8 6 (, ) (, ) Page 9

20 The graph doesequations not intersect -ais. Therefore, this equation has no solution. - Solving Linear bythe Graphing = 6 8 The related function is f () = 6 f ( ) = 6 8 f ( ) = 6( ) 8 f ( ) = 6( ) 8 f () = 6() 8 f ( 8 (, f( ) ) (, ) (, ) (, 6) f () = 6() 8 f () = 6() 8 6 The graph intersects the -ais at (, ) (, ). So the solution is Solve Algebraically + = Graph the related function, which is f ( Page

21 - Solving Linear Equations by Graphing + = Graph the related function, which is f ( The graph does not intersect the -ais. Therefore, this equation has no solution. TEXT MESSAGING Sean is sending tet messages to his friends. The function y = 6 represents the number of characters y the message can hold after he has typed characters. Find the zero and eplain what it means in the contet of this situation. To find the zero of the function, substitute zero in for y. Then y = 6 The related function is f () = f ( ) = f () = 6 f () = 6 f (9) = 6 f () = 6 f () = 6 f (6) = 6 +6 () () (9) () () (6) f ( (, f( ) ) (, 6) (, ) (9, 7) (, 6) (, ) (6, ) Page

22 - Solving Linear Equations by Graphing The graph does not intersect the -ais. Therefore, this equation has no solution. TEXT MESSAGING Sean is sending tet messages to his friends. The function y = 6 represents the number of characters y the message can hold after he has typed characters. Find the zero and eplain what it means in the contet of this situation. To find the zero of the function, substitute zero in for y. Then y = 6 The related function is f () = f ( ) = f () = 6 f () = 6 f (9) = 6 f () = 6 f () = 6 f (6) = 6 +6 () () (9) () () (6) f ( (, f( ) ) (, 6) (, ) (9, 7) (, 6) (, ) (6, ) The graph intersects the - Solve algebraically This means that the tet message is full after Sean has typed 6 characters. GIFT CARDS For her birthday Kwan receives a $5 gift card to download songs. The function m =.5d + 5 represents the amount of money m that remains on the card after a number of songs d are downloaded. Find the zero and eplain what it means in the contet of this situation. Page To find the zero of the function, substitute zero in for m. Then.

23 - Solving Linear Equations by Graphing This means that the tet message is full after Sean has typed 6 characters. GIFT CARDS For her birthday Kwan receives a $5 gift card to download songs. The function m =.5d + 5 represents the amount of money m that remains on the card after a number of songs d are downloaded. Find the zero and eplain what it means in the contet of this situation. To find the zero of the function, substitute zero in for m. Then. The related function is f (d) =.5d d f ( ) = f () = f () = f () = 6 f (6) = f () = f () =.5d + 5 f (d.5() () + 5.5() + 5.5(6) + 5.5() + 5.5() + 5 (d, f(d) ) (, 5) (, ) (, ) (6, ) (, ) (, ) The graph intersects the - Solve algebraically This means she can download a total of songs before the gift card is completely used. Solve each equation by graphing. 7 = + Page

24 - Solving Linear Equations by Graphing This means she can download a total of songs before the gift card is completely used. Solve each equation by graphing. 7 = + Rewrite with zero on the left side The related function is f () = f ( ) = + 8 f ( ) = ( ) + 8 f ( ) = ( ) + 8 f () = () + 8 f () = () + 8 f () = () + 8 f ( (, f( ) ) (, 8) (, ) (, 8) (, 6) (, ) The graph intersects the -ais at. So the solution is Solve algebraically = Rewrite with on the right side. The related function is f () = f ( ) = f ( ) = 8 f ( 8) = f ( ) = 6 ( ) 6 ( 8) 6 ( ) 6 f ( 8 (, f( ) ) (, ) ( 8, ) (, 8) Page

25 - Solving Linear Equations by Graphing = Rewrite with on the right side. The related function is f () = f ( ) = f ( ) = 8 f ( 8) = f ( ) = f () = f () = f () = 6 ( ) 6 ( 8) 6 ( ) 6 () 6 () 6 ( f ( 8 6 (, f( ) ) (, ) ( 8, ) (, 8) (, 6) (, ) (, ) The graph intersects the -ais at 8. So the solution is Solve Algebraically 5 = Rewrite 5 = with zero on the right side 5 5 The related function is f () = 5 f ( ) = f () = f () = f () = + 5 5() + 5 5() + 5 5() + 5 f ( 5 5 (, f( ) ) (, 5) (, ) (, 5) Page 5

26 - Solving Linear Equations by Graphing 5 = Rewrite 5 = with zero on the right side 5 5 The related function is f () = f ( ) = f () = f () = f () = f () = f () = f (5) = f (6) = + 5 5() + 5 5() + 5 5() + 5 5() + 5 5() + 5 5(5) + 5 5(6) + 5 f ( (, f( ) ) (, 5) (, ) (, 5) (, ) (, 5) (5, ) (6, 5) The graph intersects the - Solve algebraically = The related function is f () = f ( ) = + f ( f ( ) = ( ) + esolutions Powered by+cognero Manual f ( - ) = ( ) 7 f () = () + 6 f () = () + (, f( ) ) (, ) (, 7) (, 6) (, ) Page 6

27 - Solving Linear Equations by Graphing = The related function is f () = f ( ) = f ( ) = f ( ) = f () = f () = + ( ) + ( ) + () + () + f ( 7 6 (, f( ) ) (, ) (, 7) (, 6) (, ) f () = () + (, ) The graph intersects the -ais at. So the solution is Solve Algebraically = The related function is f () = 6 f ( ) = f ( ) = 6( ) + 5 f ( 9 (, f( ) ) (, 9) f ( ) = 6( ) + 5 f () = 6() + 5 f () = 6() + 5 f () = 6() (, ) (, 5) (, 7) (, 9) Page 7

28 - Solving Linear Equations by Graphing = The related function is f () = 6 f ( ) = f ( ) = 6( ) + 5 f ( 9 (, f( ) ) (, 9) f ( ) = 6( ) + 5 f () = 6() + 5 f () = 6() + 5 f () = 6() (, ) (, 5) (, 7) (, 9) The graph intersects the -ais at. So the solution is Solve algebraically = + The related function is f () = f ( ) = + f ( ) = ( ) + f ( ) = ( ) + f () = () + f ( 8 (, f( ) ) (, 8) 8 Page 8 (, 8) (, )

29 - Solving Linear Equations by Graphing = + The related function is f () = f ( ) = + f ( ) = ( ) + f ( 8 (, f( ) ) (, 8) f ( ) = ( ) + f () = () + f () = () + f () = () The graph intersects the -ais at (, 8) (, ) (, 6) (, 7). So the solution is Solve Algebraically = The related function is f () = f ( ) = f ( ) = ( ) f ( (, f( ) ) (, ) esolutions - Powered Cognero Manual f () = () by (, ) f () = () (, ) Page 9

30 - Solving Linear Equations by Graphing = The related function is f () = f ( ) = f ( ) = ( ) f ( (, f( ) ) (, ) f () = () (, ) f () = () f () = () (, ) (, ) The graph intersects the -ais at. So the solution is Solve algebraically 5 7 = The related function is f () = 5 f ( ) = 5 7 f ( ) = 5( ) 7 f ( ) = 5( ) 7 f () = 5() 7 f ( 67 (, f( ) ) (, 67) (, ) 7 (, 7) Page

31 - Solving Linear Equations by Graphing 5 7 = The related function is f () = 5 f ( ) = 5 7 f ( ) = 5( ) 7 f ( ) = 5( ) 7 f () = 5() 7 f ( 67 (, f( ) ) (, 67) (, ) 7 (, 7) f () = 5() 7 8 (, 8) f () = 5() 7 (, ) The graph intersects the -ais at. So the solution is Solve Algebraically The related function is Page f ( (, f( ) )

32 - Solving Linear Equations by Graphing The related function is f ( (, f( ) ) 6.5 ( 6,.5).5 (,.5).5 (,.5).5 (,.5) 6.5 (6,.5) The graph intersects the -ais at. So the solution is Solve Algebraically esolutions Manual - Powered by Cognero Solve Graphically The related function is Page

33 - Solving Linear Equations by Graphing The related function is f ( (, f( ) ).5.75 (.5,.75).5.95 (.5,.95).75 (,.75).5.55 (.5,.55).5 The graph intersects the -ais at.5 (.5,.5). So the solution is Solve Algebraically + 7 = Page

34 The graph intersects the -ais at. So the solution is - Solving Linear Equations by Graphing Solve Algebraically + 7 = The related function is f () = f ( ) = + 7 f ( ) = ( ) f ( 9) = ( 9) f ( 6) = ( 6) + 7 f ( ) = ( ) + 7 f () = () + 7 f () = () + 7 f ( (, f( ) ) (, 9) ( 9, ) ( 6, 9) (, 78) (, 7) (, 56) The graph intersects the -ais at 9. So the solution is Solve Algebraically 7 = Page

35 - Solving Linear Equations by Graphing + 7 = The related function is f () = f ( ) = + 7 f ( ) = ( ) f ( 9) = ( 9) f ( 6) = ( 6) + 7 f ( ) = ( ) + 7 f () = () + 7 f () = () + 7 f ( (, f( ) ) (, 9) ( 9, ) ( 6, 9) (, 78) (, 7) (, 56) The graph intersects the -ais at 9. So the solution is Solve Algebraically 7 = The related function is f () = f ( ) = 7 f ( f ( ) = ( ) 7 68 f ( ) = ( ) 7 f () = () 7 7 f () = () 7 esolutions Manual Powered by Cognero f () = () 7 f () = () 7 (, f( ) ) (, 68) (, ) (, 7) (, ) (, ) (, ) Page 5

36 - Solving Linear Equations by Graphing 7 = The related function is f () = f ( ) = f ( ) = ( ) f ( ) = ( ) f () = () f () = () f () = () f () = () f ( 68 7 (, f( ) ) (, 68) (, ) (, 7) (, ) (, ) (, ) The graph intersects the - Solve Algebraically SEA LEVEL Parts of New Orleans lie.5 meter below sea level. After d days of rain, the equation w =.d represents the water level w in meters. Find the zero, and eplain what it means in the contet of this situation. To find the zero of the function, substitute zero in for =.d.5.5. The related function is f () =.d d f (d) =.d.5 f ( ) =.( ).5 f ( ) =.( ).5 f () =.().5 f (d (d, f(d) ) (,.) (,.8) (,.5) Page 6

37 - Solving Linear Equations by Graphing SEA LEVEL Parts of New Orleans lie.5 meter below sea level. After d days of rain, the equation w =.d represents the water level w in meters. Find the zero, and eplain what it means in the contet of this situation. To find the zero of the function, substitute zero in for =.d.5.5. The related function is f () =.d d f (d) =.d.5 f ( ) =.( ).5 f ( ) =.( ).5 f () =.().5 f () =.().5 f (d (d, f(d) ) (,.) (,.8) (,.5) (,.) f () =.().5. (,.) The graph intersects the -ais at. So the solution is Solve Algebraically This means that the water level in New Orleans has reached sea level after about.67 days of rain. CCSS MODELING An artist completed an ice sculpture when the temperature was t=.5h shows the temperature, h hours after the sculpture s completion. If the artist completed the sculpture at 8: a.m., at what time will it begin to melt? Page 7 o To find the time the ice sculpture will melt, substitute in for t because ice melts at C.

38 - Solving Linear Equations by Graphing This means that the water level in New Orleans has reached sea level after about.67 days of rain. CCSS MODELING An artist completed an ice sculpture when the temperature was t=.5h shows the temperature, h hours after the sculpture s completion. If the artist completed the sculpture at 8: a.m., at what time will it begin to melt? o To find the time the ice sculpture will melt, substitute in for t because ice melts at C. =.5h The related function is f (h) =.5h h f (h) =.5h f ( 8) =.5( 8) f ( ) =.5( ) f () =.5() f () =.5() f (8) =.5(8) f (6) =.5(6) f (h 5 5 (h, f(h) ) ( 8, ) (, 5) (, ) (, 5) (8, ) (6, ) The graph intersects the - Solve Algebraically So, 8 hours after the sculpture was made it will start to melt. Therefore, it will start to melt at : P.M. Solve each equation by graphing. Verify your answer algebraically. 7 = 8 Page 8

39 - Solving Linear Equations by Graphing So, 8 hours after the sculpture was made it will start to melt. Therefore, it will start to melt at : P.M. Solve each equation by graphing. Verify your answer algebraically. 7 = 8 The related function is f ()= +. f ( ) = f ( ) = f ( ) = f () = f () = f () = f () = + ( ) + ( ) + () + () + () + () + f ( 5 (, f( ) ) (, 5) (, ) (, ) (, ) (, ) (, ) The graph intersects the -ais at. So, the solution is. Verify by substituting in for in the original equation. 9 + = + Manipulate the equation so that there is a zero on either side. Page 9

40 - Solving Linear Equations by Graphing 9 + = + Manipulate the equation so that there is a zero on either side. The related function is f () = + 6. f ( ) = + 6 f ( ) = ( ) + 6 f ( ) = ( ) + 6 f ( ) = ( ) + 6 f () = () + 6 f () = () + 6 f () = () + 6 f ( 6 (, f( ) ) (, ) (, ) (, ) (, 6) (, ) (, ) The graph intersects the -ais at. So, the solution is. Verify by substituting in for in the original equation = + Page The related function is f () =.

41 - Solving Linear Equations by Graphing = + The related function is f () = f ( ) = f ( ) = ( f ( ) = ( ) f () = () f () = () f () = () f () = () f ( 8. (, f( ) ) (, ) (, 8) (, ) (, ) (, ) (, ) The graph intersects the -ais at. So, the solution is. Verify by substituting in for in the original equation 5 = 5 5 The related function is f () = +. Page

42 - Solving Linear Equations by Graphing 5 = 5 5 The related function is f () = +. f ( ) = + f ( (, f( ) ) f ( ) = ( ) + (, ) f ( ) = ( ) + (, ) f () = () + (, ) f () = () + (, ) f () = () + 5 (, 5) f () = () + 6 (, 6) The graph intersects the -ais at. So, the solution is. Verify by substituting in for in the original equation. Page

43 - Solving Linear Equations by Graphing The related function is. To graph the function, make a table. f ( (, f( ) ) 5 (, 5) (, ) 5 (, 5) (, ).5 (,.5) 5 (, 5) The graph intersects the -ais at. So, the solution is. Verify by substituting in for in the original equation. Page

44 - Solving Linear Equations by Graphing The related function is. f ( (, f( ) ) (, ) (, ) 5 (, 5) From the graph, the -intercept appears to be. However, from the table above, we can see that it is not. so that you can see the - The intercept is between and., but still difficult to determine. You can find the - Page

45 - Solving by Graphing From thelinear graph,equations the -intercept appears to be. However, from the table above, we can see that it is not. so that you can see the - The intercept is between and., but still difficult to determine. You can find the - The graph intersects the -ais at. So, the solution is. Verify by substituting in for in the original equation. HAIR PRODUCTS Chemical hair straightening makes curly hair straight and smooth. The percent of the process left to complete is modeled by p =.5t +, where t is the time in minutes that the solution is left on the hair, and p represents the percent of the process left to complete. a. Find the zero of this function. b. Make a graph of this situation. c. Eplain what the zero represents in this contet. d. State the possible domain and range of this function. a. To find the zero of the function, substitute zero in for p. Page 5

46 - Solving Linear Equations by Graphing HAIR PRODUCTS Chemical hair straightening makes curly hair straight and smooth. The percent of the process left to complete is modeled by p =.5t +, where t is the time in minutes that the solution is left on the hair, and p represents the percent of the process left to complete. a. Find the zero of this function. b. Make a graph of this situation. c. Eplain what the zero represents in this contet. d. State the possible domain and range of this function. a. To find the zero of the function, substitute zero in for p. b. c. The solution must remain on the hair for 8 minutes to be completely effective. d. A possible domain is {t possible range is {p p t MUSIC DOWNLOADS In this problem, you will investigate the change between two quantities. a. Copy and complete the table. b. As the number of songs downloaded increases, how does the total cost change? c. Interpret the value of the total cost divided by the number of songs downloaded. Page 6

47 c. The solution must remain on the hair for 8 minutes to be completely effective. d. A possible domain is {t by Graphing t - Solving Linear Equations possible range is {p p MUSIC DOWNLOADS In this problem, you will investigate the change between two quantities. a. Copy and complete the table. b. As the number of songs downloaded increases, how does the total cost change? c. Interpret the value of the total cost divided by the number of songs downloaded. a. b. As the number of songs downloaded increases by the total cost increases by $. c. download a song. The cost to download a song is $. ERROR ANALYSIS Clarissa and Koko solve + 5 = + by graphing the related function. Is either of them correct? Eplain your reasoning. Page 7

48 b. As the number of songs downloaded increases by the total cost increases by $. c. - Solving Linear Equations by Graphing download a song. The cost to download a song is $. ERROR ANALYSIS Clarissa and Koko solve + 5 = + by graphing the related function. Is either of them correct? Eplain your reasoning. Koko is correct. She solved the equation so that there was a zero on one side. Then wrote the related function and graphed. Clarissa attempted to solve the equations so that there was a zero on one side. However, she added 5 to each side instead of subtracting the 5 from each. Therefore Clarissa CHALLENGE Find the solution of by graphing. Page 8

49 Koko is correct. She solved the equation so that there was a zero on one side. Then wrote the related function and - Solving Equations graphed.linear Clarissa attemptedby to Graphing solve the equations so that there was a zero on one side. However, she added 5 to each side instead of subtracting the 5 from each. Therefore Clarissa CHALLENGE Find the solution of The related function is f ()= by graphing.. Graph the related function on a graphing calculator. Verify Algebraically: CCSS TOOLS Eplain when it is better to solve an equation using algebraic methods and when it is better to solve esolutions Manual - Powered by Cognero by graphing. Page 9 It is better to solve an equation algebraically if an eact answer is needed. It is better to solve an equation graphically

50 - Solving Linear Equations by Graphing CCSS TOOLS Eplain when it is better to solve an equation using algebraic methods and when it is better to solve by graphing. It is better to solve an equation algebraically if an eact answer is needed. It is better to solve an equation graphically if an eact answer is not needed. Also, if the equation has fractions, it is easier to solve algebraically than graphically. For eample, solve graphically and OPEN ENDED Write a linear equation that has a root of. Write its related function. Thus, + =, y = +, or f () = +. WRITING IN MATH Summarize how to solve a linear equation algebraically and graphically. To solve a linear equation algebraically, solve the equation for related function by setting the equation equal to zero. Then, make a table and choose different values for and find the corresponding ycoordinate. Determine where the graph intersects the ais. This is the solution. If the graph does not intersect the ais, there is no solution. Consider the function f () = Consider the function. Solving algebraically esolutions What Manual are the - Powered and byyintercepts Cognero of the graph of the function? Page 5

51 - Solving Linear Equations by Graphing Consider the function. Solving algebraically What are the and yintercepts of the graph of the function? A, 6 B 6, C, 6 D 6, The intercept of the function is where it crosses the ais and the yvalue is zero. This point is (, ). This means that choices C and D are not options. The yintercept of the function is where it crosses the yais and the value is zero. This point is (, 6). This means that choice A is the correct option. The table shows the cost C of renting a pontoon boat for h hours. Which equation best represents the data? F C = 7.5h G C = h H C = h J C = 7.5h +.75 The difference in Cvalues is $7.5 times the difference of hvalues. This suggests C = 7.5h. So, the equation for the relationship in function notation is f () = 7.5h. The equation for choice G would be $8.5 for hour, $9.5 for hours, and $.5 for three, which does not match the data. The equation for choice H would be $.5 for hour, $7.5 for hours, and for hours, which does not match the data. The equation for choice J is $9 for hour, $6.5 for hours, and $ for three hours. So the correct choice is F. Which is the best estimate for the intercept of the graph of the linear function represented in the table? A between and B between and C between and D between and The intercept is where the linear function crosses the ais or when the yvalue is zero. This will happen between (, ) and (, ). So the best estimate for the value is between and. Therefore the correct choice is B. Page 5

52 The difference in Cvalues is $7.5 times the difference of hvalues. This suggests C = 7.5h. So, the equation for the relationship in function notation is f () = 7.5h. The equation for choice G would be $8.5 for hour, $9.5 for hours, and $.5 for three, which does not match the data. The equation for choice H would be $.5 for hour, - Solving $7.5 for Linear hours, Equations and for by hours, Graphing which does not match the data. The equation for choice J is $9 for hour, $6.5 for hours, and $ for three hours. So the correct choice is F. Which is the best estimate for the intercept of the graph of the linear function represented in the table? A between and B between and C between and D between and The intercept is where the linear function crosses the ais or when the yvalue is zero. This will happen between (, ) and (, ). So the best estimate for the value is between and. Therefore the correct choice is B. EXTENDED RESPONSE Mr. Kauffmann has the following options for a backyard pool. If each pool has the same depth, which pool would give the greatest area to swim? Eplain your reasoning. Find the area of each pool. The area of the rectangular pool is found by: So, the area of the rectangular pool is 6 ft. The area of the circular pool is found by: So, the area of the circular pool is about 5.5 ft. Finally, the area of the rounded pool can be found by splitting it into a circle and a rectangle. So, it can be found by: esolutions Manual - Powered by Cognero Page 5

53 The intercept is where the linear function crosses the ais or when the yvalue is zero. This will happen - Solving between Linear (, ) and Equations (, ). So by the Graphing best estimate for the value is between and. Therefore the correct choice is B. EXTENDED RESPONSE Mr. Kauffmann has the following options for a backyard pool. If each pool has the same depth, which pool would give the greatest area to swim? Eplain your reasoning. Find the area of each pool. The area of the rectangular pool is found by: So, the area of the rectangular pool is 6 ft. The area of the circular pool is found by: So, the area of the circular pool is about 5.5 ft. Finally, the area of the rounded pool can be found by splitting it into a circle and a rectangle. So, it can be found by: So the area of the rounded pool is about. ft. Therefore, the circular pool would have the greatest area. Find the and yintercepts of the graph of each linear equation. y = + To find the intercept, let y =. Page 5

54 - Solving the area Linear of the Equations rounded pool by Graphing is about. ft. Therefore, the circular pool would have the greatest area. Find the and yintercepts of the graph of each linear equation. y = + To find the intercept, let y =. To find the yintercept, let =. So, the intercept is 5 and the yintercept is. y = 6 9 To find the intercept, let y =. To find the yintercept, let =. Page 5

55 - Solving So, the intercept Linear Equations is 5 and by the Graphing yintercept is. y = 6 9 To find the intercept, let y =. To find the yintercept, let =. So, the intercept is, and the yintercept is. y = 8 To find the intercept, let y =. To find the yintercept, let =. Page 55

56 - Solving So, the intercept Linear Equations is, and by the Graphing yintercept is. y = 8 To find the intercept, let y =. To find the yintercept, let =. So, the intercept is 7, and the yintercept is. FOOD If % milk contains % butterfat and whipping cream contains 9% butterfat, how much whipping cream and % milk should be mied to obtain 5 gallons of milk with % butterfat? Let m represent the gallons of milk needed and let w represent the gallons of whipping cream needed. m gallons of milk with % butterfat must be combined with w gallons of whipping cream with 9% butterfat to get a 5-gallon miture with % butterfat. amount % butter fat % butter fat milk m..m whipping w.9.9w cream 5..(5) The equation would be,.m +.9w = (.)(5). Substitute 5 w for m w) +.9w = (.)(5). Page 56

57 - Solving Linear Equations by Graphing So, the intercept is 7, and the yintercept is. FOOD If % milk contains % butterfat and whipping cream contains 9% butterfat, how much whipping cream and % milk should be mied to obtain 5 gallons of milk with % butterfat? Let m represent the gallons of milk needed and let w represent the gallons of whipping cream needed. m gallons of milk with % butterfat must be combined with w gallons of whipping cream with 9% butterfat to get a 5-gallon miture with % butterfat. amount % butter fat % butter fat milk m..m whipping w.9.9w cream 5..(5) The equation would be,.m +.9w = (.)(5). Substitute 5 w for m w) +.9w = (.)(5). To find the gallons of milk, substitute for w in the original equation: m = 5 = 5. So, you need gallons of whipping cream and 5 gallons of % milk to make a 5-gallon miture that has % butterfat. Translate each sentence into an equation. The product of and m plus times n is the same as the quotient of and p. The sum of and five times y equals twice z minus 7. Simplify. Page 57

58 - Solving Linear Equations by Graphing Evaluate a = 6, b =, c = 9, d = a = 8, b =, c = 5, d = a =, b = 7, c =, d = Page 58

59 - Solving Linear Equations by Graphing a =, b = 7, c =, d = Page 59

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