1. 12345678 -8-7-6-5-4-3-2-1 8 7 6 5 4 3 2 1 - 8 - 7 - 6 - 5 - 4 - 3 - 2 - 1 x y The Rectangular Coordinate System M Bartlett 2002 Slide 3.1

2. 12345678 -8-7-6-5-4-3-2-1 8 7 6 5 4 3 2 1 - 8 - 7 - 6 - 5 - 4 - 3 - 2 - 1 x y QuadrantI QuadrantII QuadrantIII QuadrantIV Origin M Bartlett 2002 Slide 3.2 3. 12345678 -8-7-6-5-4-3-2-1 8 7 6 5 4 3 2 1 - 8 - 7 - 6 - 5 - 4 - 3 - 2 - 1 x y P (3, 3) Q ( 4, 6) R (6, 4) M Bartlett 2002 Slide 3.3 4. ( 4, 6)( 2, 5)(0, 4)(2, 3)(4, 2) M Bartlett 2002 Slide 3.4 12345678 -8-7-6-5-4-3-2-1 8 7 6 5 4 3 2 1 - 8 - 7 - 6 - 5 - 4 - 3 - 2 - 1 x y (4, 2) 2 4 (2, 3) 3 2 (0, 4) 4 0 ( 2, 5) 5 2 ( 4, 6) 6 4 ( x,y ) y x 5. ( 4, 9)( 2, 6)(0, 3 )(2, 0)(4, 3) M Bartlett 2002 Slide 3.5 12345678 -8-7-6-5-4-3-2-1 8 7 6 5 4 3 2 1 - 8 - 7 - 6 - 5 - 4 - 3 - 2 - 1 x y (4, 3) 3 4 (2, 0) 0 2 (0, 3) 3 0 ( 2, 6) 6 2 ( 4, 9) 94 ( x,y ) y x 6. (4, 0) (0, 1 ) ( 4, 2) M Bartlett 2002 Slide 3.6 12345678 -8-7-6-5-4-3-2-1 8 7 6 5 4 3 2 1 - 8 - 7 - 6 - 5 - 4 - 3 - 2 - 1 x y (4, 0) 0 4 (2,) 2 (0, 1) 1 0 ( 2,) 2 ( 4, 2) 2 4 ( x,y ) y x (2,) (2,) 7. ( 3, 2)( 1, 2)(0, 2 )(2, 2)(4, 2) M Bartlett 2002 Slide 3.7 12345678 -8-7-6-5-4-3-2-1 8 7 6 5 4 3 2 1 - 8 - 7 - 6 - 5 - 4 - 3 - 2 - 1 x y (4, 2) 2 4 (2, 2) 2 2 (0, 2) 2 0 ( 1, 2) 2 1 ( 3, 2) 2 3 ( x,y ) y x 8. ( 3, 4)( 3, 2)( 3, 0)( 3, 2)( 3, 4) M Bartlett 2002 Slide 3.8 12345678 -8-7-6-5-4-3-2-1 8 7 6 5 4 3 2 1 - 8 - 7 - 6 - 5 - 4 - 3 - 2 - 1 x y ( 3, 4) 4 3 ( 3, 2) 2 3 ( 3, 0) 0 3 ( 3, 2) 2 3 ( 3, 4) 4 3 ( x,y ) y x 9. APPLICATION

Example 1:

A car purchased for $17,000 is expected to depreciate according to the formula

y= 1,360 x+ 17,000. When will the car be worthless?

M Bartlett 2002 Slide 3.9 Solution : The car is worthless wheny= 0. 1360 x+ 17000 = 0 17000 = 1360 x +1360 x +1360 x x= 12.5 The car will be worthless in 12.5 years. 10. The Distance Formula d | x 2x 1 | | y 2y 1 | x y M Bartlett 2002 Slide 3.10 11. The Midpoint Formula M Bartlett 2002 Slide 3.11 x y 12. Slope of a Line Suppose that a college student rents a room for $300 per month, plus a $200 non-refundable deposit. Construct a table that shows the cost ( y ) for different numbers of months ( x ). Construct a graph from this data. M Bartlett 2002 Slide 3.12 1,400 4 1,100 3 800 2 500 1 200 0 Total Cost ( y ) Time in Months ( x ) 13. The Slope of a Non-vertical Line x 2x 1 y 2y 1 M Bartlett 2002 Slide 3.13 x y 14. EXAMPLE 1 :Find the slope of the line passing through P(1, 2) and Q(7, 8). P (1, 2) Q (7, 8) Run = 8 R (7, 2) Rise = 10 M Bartlett 2002 Slide 3.14 12345678 -8-7-6-5-4-3-2-1 8 7 6 5 4 3 2 1 - 8 - 7 - 6 - 5 - 4 - 3 - 2 - 1 x y 15. EXAMPLE 2 :Find the slope of the line determined by 3x 2y =9. P (0, 4.5) Q (3, 0) R (3, 4.5) x= 0 ,y= 4.5 y= 0,x= 3 Rise = 4.5 Run = 3 M Bartlett 2002 Slide 3.15 12345678 -8-7-6-5-4-3-2-1 8 7 6 5 4 3 2 1 - 8 - 7 - 6 - 5 - 4 - 3 - 2 - 1 x y 16. Exampe 3 :If carpet cost $25 per square yard plus a delivery charge of $30, the total costcofnyards is given by the formula Total cost equals cost per square yard times The number of square yards purchased plus the delivery charge M Bartlett 2002 Slide 3.16 17. Slope = 0 Slope undefined M Bartlett 2002 Slide 3.17 x y x y 18. Positive Slope Negative Slope M Bartlett 2002 Slide 3.18 x y x y 19. The Slope of Parallel Lines A C B Slope =m 1 D FE Slope =m 2 M Bartlett 2002 Slide 3.19 x y 20. EXAMPLE 4 :The lines in the figure below are parallel. Find x.R(2, 5) T (x , 0) Q(3, 4) P(1, 2) Slope ofPQ= slope ofRT M Bartlett 2002 Slide 3.20 12345678 -8-7-6-5-4-3-2-1 8 7 6 5 4 3 2 1 - 8 - 7 - 6 - 5 - 4 - 3 - 2 - 1 x y 21. The Slope of Perpendicular Lines P ( a ,b ) Slope =m 1 O (0, 0) Q ( c ,d ) Slope =m 2 M Bartlett 2002 Slide 3.21 x y 22. EXAMPLE 5 :Are the lines shown in the figure below perpendicular?Q(9, 4) O (0, 0) P(3, 4) M Bartlett 2002 Slide 3.22 12345678 -8-7-6-5-4-3-2-1 8 7 6 5 4 3 2 1 - 8 - 7 - 6 - 5 - 4 - 3 - 2 - 1 x y 23. Point Slope of the Equation of a Line x=xx 1 M Bartlett 2002 Slide 3.23 Slope =m y=yy 1 yy 1=m ( xx 1 ) x y 24. EXAMPLE 5 :Write the equation of the line with slopepassing through P(4, 5). P (4, 5) M Bartlett 2002 Slide 3.24 Q ( x ,y ) Run = 3 Rise = 212345678 -8-7-6-5-4-3-2-1 8 7 6 5 4 3 2 1 - 8 - 7 - 6 - 5 - 4 - 3 - 2 - 1 x y 25. EXAMPLE 6 :Find the equation of the line passing through P(5, 4) and Q(8, 6). P (5, 4) Run = 13 Q (8, 6) Rise = -10 M Bartlett 2002 Slide 3.25 12345678 -8-7-6-5-4-3-2-1 8 7 6 5 4 3 2 1 - 8 - 7 - 6 - 5 - 4 - 3 - 2 - 1 x y 26. Slope Intercept Form of the Equation of a Line M Bartlett 2002 Slide 3.26 Slope =m y- intercept x y 27. Graphing Equations Written in Slope Intercept Form M Bartlett 2002 Slide 3.27 P (0, 2) Q (3, 2) x=3 y=4 12345678 -8-7-6-5-4-3-2-1 8 7 6 5 4 3 2 1 - 8 - 7 - 6 - 5 - 4 - 3 - 2 - 1 x y 28. Example 7 :Find the slope and the y intercept of the line with equation 2( x 3) = 3( y+ 5). Then graph it. 2 x 6 = 3 y 153 y 6 = 2 x 153 y = 2 x 9y intercept is (0, 3) 3 2(0, 3) (3, 5) 2( x 3)= 3( y+ 5)Slide 3.28 M Bartlett 2002 12345678 -8-7-6-5-4-3-2-1 8 7 6 5 4 3 2 1 - 8 - 7 - 6 - 5 - 4 - 3 - 2 - 1 x y 29. Example 8 :Are the lines represented by the equationsy= 3 x+ 2 and 6 x+ 2 y= 5 parallel? Example 9 :Are the lines represented by the equations 2 x+ 3 y= 9 and 3 x 2 y= 5 perpendicular? Example 10 :Write the equation of the line passing throughP (1, 2) and parallel to the liney= 8 x 3. Example 11 :Write the equation of the line passing throughQ (1, 2) and perpendicular to the liney= 8 x 3. Slide 3.29 M Bartlett 2002 30. General Form of the Equation of a Line IfA ,B , andCare real numbers andB 0, the graph ofAx+By=C(General Form of the equation of a line.) is a non-vertical line with slope ofand ay intercept of.IfB= 0, the graph is a vertical line withx intercept of. Slide 3.30 M Bartlett 2002 31. Example 12 :Find the slope andy intercept of the graph of 3 x 4 y= 12. Slide 3.31 M Bartlett 2002 Solution: Step 3:Step 4:Step 1:Step 2: 32. Summary

General FormA x +B y =C

AandBcannot both be 0.

Slope intercept formy=mx+b

The slopem , and they -intercept is (0,b ).

Point slope formyy 1=m ( xx 1 )

The slope ism , and the line passesthrough ( x 1 ,y 1 ).

A horizontal liney=b

The slope is 0, and theyintercept is (0,b ).

A vertical Linex=a

There is no defined slope, and thex -intercept is ( a , 0).

Slide 3.32 M Bartlett 2002 33. Example 13 :A taxi cab was purchased for $24,300. Its salvage value at the end of its 7-year useful life is expected to be $1,900. Ifyis the value of the taxi cab afterx -years of use, andyandxare related by the equation of a line, a. Find the equation of the line, called thedepreciation equation .b. Find the value of the taxi cab after 3 years.c. Find the economic meaning of they -intercept ofthe line. d. Find the economic meaningof the slope of theline.Slide 3.33 M Bartlett 2002 34. M Bartlett 2002 Slide 3.34 Intercepts of Graphs x -intercepts y -intercept x y x y 35. Graphing an Equation M Bartlett 2002 Slide 3.1 Slide 3.35 (-3, 5) (-2, 0) (-1, -3) (1, -3) (2, 0) (3, 5) V (0, -4) This is an example of aParabola . The lowest point being the vertexV (0, 4). Since they axis divides the graph into two congruenthalves it is called the axis of symmetry . The parabola issymmetric about they axis . 12345678 -8-7-6-5-4-3-2-1 8 7 6 5 4 3 2 1 - 8 - 7 - 6 - 5 - 4 - 3 - 2 - 1 x y (3, 5) 5 3 (2, 0) 0 2 (1, -3) -3 1 (0, -4) -4 0 (-1, -3) -3 -1 (-2, 0) 0 -2 (-3, 5) 5 -3 ( x ,y ) y x 36. M Bartlett 2002 Slide 3.36 Symmetries of Graphs y axis symmetry Originsymmetry x axis symmetry x y x y x y 37. Absolute Value Graph M Bartlett 2002 Slide 3.1 Slide 3.37 (1, 1) (2, 2) (0, 0) (3, 3) (4, 4) 12345678 -8-7-6-5-4-3-2-1 8 7 6 5 4 3 2 1 - 8 - 7 - 6 - 5 - 4 - 3 - 2 - 1 x y (4, 4) 4 4 (3, 3) 3 3 (2, 2) 2 2 (1, 1) 1 1 (0, 0) 0 0 ( x ,y ) y x 38. Example 14: Graphy=x 3 9 x . (-3, 0) (-2, 10) (-1, 8) (0, 0) (1, -8) (2, -10) (3, 0) M Bartlett 2002 Slide 3.38 y x (3, 0) 27 3 (2, -10) 8 2 (1, -8) -8 1 (0, 0) 0 0 (-1, 8) 8 -1 (-2, 10) 10 -2 (-3, 0) 0 -3 ( x ,y ) y x 39. M Bartlett 2002 Slide 3.1 Slide 3.39 (1, -1) (4, -2) (0, 0) (9, -3) 12345678 -8-7-6-5-4-3-2-1 8 7 6 5 4 3 2 1 - 8 - 7 - 6 - 5 - 4 - 3 - 2 - 1 x y (9, -3) -3 9 (4, -2) -2 4 (1, -1) -1 1 (0, 0) 0 0 ( x ,y ) y x Example 15: Graph 40. M Bartlett 2002 Slide 3.1 Slide 3.40 (1, 1) (4, 2) (0, 0) (9, 3) Example 16: Graphy 2=x . 12345678 -8-7-6-5-4-3-2-1 8 7 6 5 4 3 2 1 - 8 - 7 - 6 - 5 - 4 - 3 - 2 - 1 x y (9, 3) 3 9 (4, 2) 2 4 (1, 1) 1 1 (0, 0) 0 0 ( x ,y ) y x 41. Circles Acircleis the set of all points in a plane that are a fixed distance from a point called thecenter . The fixed distance is called theradius of the circle . M Bartlett 2002 Slide 3.41 42. Standard Equation of a Circle C( h ,k ) P( x ,y ) r M Bartlett 2002 Slide 3.42 y x 43. The Standard Equation of a Circle with Center at ( h ,k ) The graph of any equation that can be written in the form ( xh ) 2+ ( yk ) 2=r 2 is a circle of radiusrand center at point ( h ,k ). M Bartlett 2002 Slide 3.43 44. The Standard Equation of a Circle with Center at (0, 0) The graph of any equation that can be written in the form x 2+y 2=r 2 is a circle of radiusrand center at the origin. The general form of the equation of a circle is given by x 2+y 2+cx+dy+e= 0 wherec ,d , andeare real numbers. M Bartlett 2002 Slide 3.44 45. Example 17 :Find the general form of the equation of the circle with radius 6 and center at ( 2, 5).Slide 3.45 M Bartlett 2002 Solution :( xh ) 2+ ( yk ) 2=r 2 46. Example 18 :Find the general form of the equation of a circle with endpoints of its diameter at ( 2, 2) and (6, 8).Slide 3.46 M Bartlett 2002 47. Graphing Equations of Circles M Bartlett 2002 Slide 3.1 Slide 3.47 (-1, 2) Example 19 :Graph 2 x 2+ 2 y 2+ 4 x 8 y= 2.Radius 2 12345678 -8-7-6-5-4-3-2-1 8 7 6 5 4 3 2 1 - 8 - 7 - 6 - 5 - 4 - 3 - 2 - 1 x y 48. Proportions is called a Proportion. The numbersaanddare called theextremes . The numbersbandcare called themeans . M Bartlett 2002 Slide 3.48 49. Property of Proportions M Bartlett 2002 Slide 3.49 In any proportion, the product of the extremes isequal to the product of the means. Example 21 : The ratio of women to men in amathematics class is 3 to 5. How many womenare in the class if there are 30 men?Example 20 : Solve the proportion 50. Direct Variation The words yvaries directly withx , or yisdirectly proportional tox , mean thaty=kxfor some real number constantk . The numberkis called theconstant ofproportionality. M Bartlett 2002 Slide 3.50 m= 2 m= 1 y x 51. Example 22 :The distance that an object will fall intseconds varies directly with the square oft . An object falls 16 feet in 1 second. How long will it take to fall 144 feet?Slide 3.51 M Bartlett 2002 52. Inverse Variation M Bartlett 2002 Slide 3.52 The words yvaries inversely withx , or yisinversely proportional tox , mean thaty=k/xorxy=kfor some real number constantk . (1, 1) (3, 3) (2, -2) In each case, the equation determines on branch of a curve called aHyperbola . y x (c) y x (a) y x (b) 53. Example 23 :Intensity of illumination from a light source varies inversely with the square of the distance from the source. If the intensity of the light source is 100 lumens at a distance of 20 feet, find the intensity at 30 feet.Slide 3.53 M Bartlett 2002 54. Joint Variation M Bartlett 2002 Slide 3.54 The words yvaries jointly withwandx meanthaty=kwxfor some real number constantk . 55. Example 24 :The area of a rectangle varies jointly with its length and width. Find the constant of proportionality.Slide 3.55 M Bartlett 2002 56. Combined Variation M Bartlett 2002 Slide 3.56 yvaries directly withxand inversely withz . yvaries jointly with the square ofxand the cube root ofz . yvaries jointly withxand the square root ofz and inversely with the cube root oft . yvaries inversely with the product ofxandz . 57. Example 25 :The power, in watts, dissipated as heat in a resistor varies directly with the square of the voltage and inversely with the resistance. If 20 volts are placed across a 20-ohm resistor, it will dissipate 20 watts. What voltage across a 10-ohm resistor will dissipate 40 watts?Slide 3.57 M Bartlett 2002