2.1 The Rectangular Coordinate System


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1 . The Rectangular Coordinate Sstem In this section ou will learn to: plot points in a rectangular coordinate sstem understand basic functions of the graphing calculator graph equations b generating a table of values graph equations using  and intercepts Understanding the Rectangular Coordinate Sstem: points/ordered pairs origin  and aes quadrants For this course ou must be able to use our graphing calculator to perform the following functions:. enter equations (You must be able to solve the equation for.). generate a table of values. determine an appropriate viewing rectangle (window). graph equations using an appropriate window Eample : Generate a table of values to graph the equations below without using a calculator. Then check the table values and graph using a graphing calculator. (a) (b) Page (Section.)
2 The standard viewing rectangle or viewing window for most calculators is [0, 0, ] b [0, 0, ] or [minimum value, maimum value, ais scale] b [minimum value, maimum value, ais scale] determined b the  and aes. The viewing rectangle for the graph in Eample is. Eample : Draw a viewing rectangle to represent [8, 0, ] b [0,, ]. The intercept of a graph is the coordinate of a point where the graph intersects the ais. To find the intercept:. Substitute 0 for value.. Solve for. The intercept of a graph is the coordinate of a point where the graph intersects the ais. To find the intercept:. Substitute 0 for value.. Solve for. Eample : Graph the equation using intercepts. 8 8 Page (Section.)
3 . Homework Problems A. Refer to the graph at right to determine the coordinates of points A F.. Determine in which quadrant(s) or on which ais the point (, ) must lie based on the following conditions: (a) > 0 and < 0 (b) < 0 (c) > 0 (d) < 0 and 0 8 < E C D 8 F B (e) < 0 and > 0 (f) > 0and 0 (g) > 0 and < 0 (h) < 0 and 0. Complete the table of values for to find coordinates (, ) Given the equation, find the values for each of the ordered pairs: (, ), (, ), (, ), (0, ), (, ), (, ), (, ). Find the  and intercepts of the graphs for each of the equations. (a) + 0 (b) (c) (d) + 0 (e) ( + ). A car purchased for $8,0 is epected to depreciate according to the formula , where is the value after ears. When will the car no longer have an value?. Homework Answers:. A(, ); B(, ); C(, 0); D(, ); E(0, ); F(, ). (a) IV; (b) III; (c) I or III; (d) II or IV; (e) II; (f) positive ais; (g) III; (h) negative ais. (, ); (, ); (, ); (0, ); (, ); (, ); (, ). (, ); (, 0); (, ); (0, ); (,); (, 0); (, ) 0. (a) (, 0) and (0, ); (b) (, 0) and (0, 8); (c) (, 0) and 0, ; (d), 0 9 (e),0 and (0, ). in. ears Page (Section.) and (0, );
4 . Slope and Average Rate of Change In this section ou will learn to: find the slope of an oblique (slanted) line find the slope of horizontal and vertical lines find the average rate of change Definition: The slope of the line through the distinct points, ) ( and, ) is ( change in change in or rise run or or or, where. Eample : Find the slope of the line containing the following points: (a) (, ) and (, ) (b) (, ) and (8, ) (c) (, ) and (, 8) Positive Slope Negative Slope Zero Slope Undefined Slope m > 0 m < 0 m 0 m is undefined Page (Section.)
5 If a graph is not a straight line, the average rate of change between an two points is the slope of the line containing the two points. This line is called a secant line. Let (, ) and (, ) be distinct points on a graph. The average rate of change from to is where Eample : The minimum wage in 9 was $.0. The minimum wage in 009 was $.. Find the average rate of change in the minimum wage from 9 to 009. Round to nearest cent. Eample : Find the average rate of change on the graph of ( ) from to. 8 Eample : Refer to the graph below to find the average rate of change (ARC) of the blood alcohol level to 0 hours after drinking. What does this represent? Page (Section.)
6 . Homework Problems. Find the slope of the line passing through each pair of points or state that the slope is undefined. (a) (, ) and (, ) (b) (, ) and (, ) (c), and, (d) (8, ) and (8, ) (e) (0, 0) and (, ) (f) (, b) and (, b) (g) (0, b) and (a, a + b) where a 0 (h) (a + b, c) and (b + c, a) where c a. Refer to the federal minimum wage rates in the table below to determine the average rate of change in the minimum wage for the given time periods. (Round to nearest cent.) Year Minimum Wage $.. $.0 $.0 $. $. $. $. (a) 9 to 0 (b) 9 to 9 (c) 9 to 0 (d) 989 to 99 (e) 009 to 0. Find the average rate of change on the graph of ( + ) from (a) to (b) to (c) to 8 (d) 8 to 0. Find the average rate of change on the graph of + from to.. Refer to the graph to find the average rate of change from (a) to (b) to (c) to (d) to. Homework Answers:. (a) ; (b) ; (c) ; (d) undefined; (e) ; (f) 0; (g) ; (h) . (a) $./ear; (b) $.0/ear; (c) $./ear (d) $. or $./ear; (e) $0.00/ear. (a) ; (b) ; (c) ; (d) (a) ; (b) 0; (c) ; (d) Page (Section.)
7 . Writing Equations of Lines In this section ou will learn to use pointslope form to write an equation of a line use slopeintercept form to write an equation of a line graph linear equations using the slope and intercept find the slopes and equations of parallel and perpendicular lines recognize and use the standard form of a line PointSlope Form The pointslope form of the equation of a nonvertical line with slope m that passes through, ) is ( m( ) SlopeIntercept Form The slopeintercept form of the equation of a nonvertical line with slope m and intercept b is m + b Eample : Find an equation for the line that passes through the point (, ) and has a slope equal to . Write the equation in pointslope form. Then write the equation in slopeintercept form. (Solve for.) Steps:. Substitute, and m values.,. Simplif and solve for.. Check given point and slope (m). Eample : Find an equation for the line passing through the points (, 8) and (, ). Write our equation in pointslope form and then in slopeintercept form. Steps:. Find the slope.. Substitute the slope and the values for one of the points.. Simplif and solve for.. Check the point and slope. Page (Section.)
8 Eample (Optional): Find an equation for the line passing through (, ) and m using the PointSlope Method SlopeIntercept Method Eample : Graph each of the following equations. (a) + and + (b) and Steps: Plot the intercept.. Use the slope rise m run to find a nd point.. Draw a line through the points. (c) f ( ) (d) and Horizontal Line Equations: Page (Section.) Vertical Line Equations:
9 General Form of the Equation of a Line: Ever line has an equation that can be written in the A + B + C general for 0, where A, B, and C are real numbers, and A and B are not both zero. (Note: Solve the equation for to find the slope and intercept.) Eample : Find the slope and the intercept for the line whose equation is Eample : Find the slope and the intercept for the line whose equation is A + B + C 0 Parallel Lines Slopes are equal.* m m Vertical lines (undefined slopes) are parallel. *If the lines are not vertical lines. Perpendicular Lines Slopes are negative reciprocals.* The product of their slopes is .* m m A horizontal line with slope 0 is perpendicular to a vertical line with an undefined slope. Eample : Find an equation for the line through the point (, ) and parallel to the line whose equation is 0. Write the equation in slopeintercept form. Page (Section.)
10 Eample 8: Complete the table below for the perpendicular lines l and l. Slope of l undefined Slope of l 0. Eample 9: Determine whether the graphs of the equations below are parallel, perpendicular or neither. (a) and (b) + 0 and + Eample 0: Find an equation for the line passing through (, ) and perpendicular to 0. Eample : Find an equation for the line passing through the point (, ) and perpendicular to the graph of the line 0. Page (Section.)
11 . Homework Problems:. Find an equation for each line based on the conditions below. Write the equation in slopeintercept form and also standard form. (a) passing through (, ); m  (b) passing through (, ); m (c) intercept (, 0); m (d) passing through (, ); m (e) passing through (, ); m 0 (f) passing through (8, ); slope is undefined. Write an equation for the line that passes through the two points. Write the answer in slopeintercept form. (a) (, ) and (, ) (b) (, 0) and (, 8) (c) (, ) and (, ) (d), and, (e) (,.) and (, ) (f) (, a) and (, a). Find an equation for the line that has the following intercepts. Write the equation in standard form. (a) (0, ) and (, 0) l (b) ( 0, ) and, 0. Find the equations for the lines l l on the graph at the right.. Find the slope and intercept for each of the lines below. l 8 (a) + (b) 8 + (c) + ( + ) (d) B C + A. Determine whether the lines are parallel, perpendicular, or neither. (a) + and (b) and (c) + and (d) + and. Use the given conditions to write an equation for each line in slopeintercept form. l 8 (a) passing through (8, 0) and parallel to the line whose equation is + (b) passing through (, ) and perpendicular to the line whose equation is + 0 Page (Section.)
12 (c) passing through (, ) and parallel to the line whose equation is (d) passing through (, ) and perpendicular to the line whose equation is (e) passing through (, ) and perpendicular to (f) passing through (, ) and is perpendicular to the line with an intercept of and a intercept of  (g) perpendicular to the line whose equation is 0 and has the same intercept as this line 8. Find the and values if the line through the given points has the indicated slope. (a) (, ), (, ), and (, ); m (b) (, 9), (, ), and (, ); m 9. Find the coefficients a and b for the equation a + b 0 so that the graph of the line will have an intercept of and a intercept of . (Use the definition of intercepts to find a and b.) 0. The minimum wage at ABC Department Store in 9 was $.. The minimum wage for this store in 00 was $.. (Note: Round all values for this problem to nearest hundredths.) (a) Use this information to find the equation of the line that models this data in pointslope form. (b) Use this information to find the equation of the line that models this data in slopeintercept form. (c) Use our model to predict the minimum wage for 0. (d) What is the average rate of change in minimum wage from 9 to 00?. Homework Answers:. (a) 0 ; ; (b) + ; + 0 ; (c) + ; ; + 0; (e) ; 0 ; (f) 8 0. (a) +; (b) + ; (c) ; (d) ; (e) + ; (f) 8 8 a. (a) + 0 ; (b). l : + ; l : ; l. (a) ; B A (b) ; (c) 0; (d) m ; 8 C C. (a) parallel; (b) perpendicular; (c) neither; (d) perpendicular. (a) ; (b) + ; (c) + ; (d) +; (e) ; (f) + ; (g) 8. (a) ; ; (b) ; 0 9. a ; b 0 0. (a) (.).( 9) or (.).( 00) ; (b). 9. ; (c) $8.; (d) $./ear Page (Section.)
13 . Proportion & Variation In this section ou will learn to solve proportions use direct variation to solve problems use indirect variation to solve problems use joint variation to solve problems use mied variation to solve problems A proportion is an equation that indicates that two ratios are equal. Eample: 0,, 8 8 0, or ( + ) a c In the proportion, a and d are called the etremes, and b and c are called the means. The product b d of the means the product of the etremes: ad bc. This is sometimes referred to as the cross product. Note: The ratio of to or can also be epressed as :. Eample : Solve the proportion: Eample : If gasoline and oil are to be mied in a 0: ratio for a Stihl chainsaw, how man ounces of oil should be mied with gallons of gasoline? (There are 8 ounces in a gallon.) Page (Section.)
14 Direct Variation: Given two quantities and, varies directl with or is directl proportional to, if there is a (nonzero) constant k such that k. (k is called the constant of variation or constant of proportion.) Note: In a direct variation, the ratio of the variables is a constant: k (or k where k is a constant). Eample : Write a variation equation for each statement below. Distance driven varies directl with the speed (rate) of a vehicle. The circumference of a circle varies directl with the length of its diameter. The distance that a bod falls from rest varies directl as the square of the time. Cost varies directl with the purchased quantit. Eample : The volume of blood, B, in a human bod varies directl as the weight, W, of the bod. A person who weighs 0 pounds has approimatel quarts of blood. Find the constant of variation and then write a variation equation. Estimate the number of quarts of blood in a person who weighs 0 pounds. Now solve this problem using ratios/proportions. Eample : Suppose varies directl as and when. Find when. Page (Section.)
15 Indirect Variation: Given two quantities and, varies inversel with or is inversel proportional to, if there is a (nonzero) constant k such that k k or. Note: In an indirect variation, the product of the variables is a constant: k (where k is a constant). Eample : Write a variation equation for each statement below. In a closed container, pressure varies inversel with the volume of gas. The intensit of light varies inversel with the square of the distance from the source. c varies inversel with the sum of a and b. The force of gravit varies inversel as the square of the distance between the objects. Eample : (Bole s Law) In a closed container with constant temperature, the pressure of a gas varies inversel with its volume. Suppose the steam in the clinder of an historic locomotive eerts a pressure of 00 pounds per square inch (psi) when the volume of the clinder is cubic inches. Find the constant of variation and then write the variation equation. Use our variation equation to determine the pressure when the return stroke of the piston increased the volume to cubic inches. Eample 8: varies inversel as the square root of and when. Find when 00. Page (Section.)
16 Joint Variation is a variation in which a variable varies directl as the product of two or more variables. For eample, varies jointl as and z can be epressed as kz. Combined Variation is a variation which involves both direct (or joint) and indirect variation. For eample, varies jointl as and z and inversel with the square of w can be epressed as z kz kz, k, or w w w Eample 8: Write a variation equation for each statement below. Interest earned varies jointl with the rate of interest and the length of time on deposit. Horsepower varies jointl as the number of clinders in the engine and the square of the diameter of the clinder. The electrical resistance in a wire varies directl with its length and inversel as the crosssectional area of the wire. Eample 9: The volume of a cone, V, varies jointl as its height, h, and the square of its radius, r. A cone with a radius measuring feet and a height measuring 0 feet has a volume of 0π cubic feet. Find the volume of a cone having a radius of feet and a height of feet. Eample 0: Bodmass inde, or BMI, takes both weight and height into account when assessing whether an individual is underweight or overweight. BMI varies directl as bod weight, in pounds, and inversel as the square of bod height, in inches. A person who weighs 80 pounds and is feet tall has a BMI of.. What is the BMI for a 0pound person who is feet 0 inches tall? (Normal values for the BMI in adults are between 0 and. Values below 0 indicate that an individual is underweight and values above 0 indicate that an individual is obese.) Page (Section.)
17 . Homework Problems:. Solve each proportion. (a) (b) 9 8 (c) miles per hour is equivalent to 88 kilometers per hour. If a speed limit is posted at 80 kilometers per hour, what is the speed limit in miles per hour (rounded to the nearest mile)?. Western High School has 8 teachers. If the high school has a studenttoteacher ratio of :, how man students attend Wester High School?. A foot flagpole casts a shadow 0 feet long. A nearb office building casts a shadow feet long. How tall is the building?. Write a variation equation for each statement below. (a) varies jointl as and z. (b) varies directl as the cube root of z and inversel as. (c) varies jointl as and z and inversel as the square of w. (d) varies directl as z and inversel as the sum of and w.. varies directl as. when. Find when 8.. varies inversel as. when. Find when. 8. a varies directl as b and inversel as the square of c. a when b 9 and c. Find a when b and c C varies jointl as A and T. C when A 00 and T. Find C when A 00 and T. 0. An object s weight on the moon, M, varies directl as its weight on Earth. An astronaut weighs 0 pounds with all of his equipment on Earth and 0 pounds on the moon. What would a 8pound person on Earth weigh on the moon?. The range of a projectile varies directl with the square of its initial velocit. In a circus act, a performer is shot out of a cannon with an initial velocit of 80 feet per second into a net 00 feet awa. How far awa should the net be placed if the initial velocit is increased to 9 feet per second?. The time it takes for a pendulum to complete one period (swing back and forth) varies directl as the square root of its length. If a pendulum 0 feet long has a period of seconds, find the period (to the nearest tenth of a second) of a pendulum 0 feet long. Page (Section.)
18 . Kinetic energ varies jointl as the mass and the square of the velocit. A mass of 8 grams and velocit of centimeters per second has a kinetic energ of ergs. Find the kinetic energ for a mass of grams and velocit of centimeters per second.. Sound intensit varies inversel as the square of the distance from the sound source. If ou are in a movie theater and ou change our seat to one that is twice as far from the speakers, how does the new sound intensit compare to that of our original seat?. The time it takes to build a highwa varies directl with the length of the road but inversel with the number of workers. If it takes 00 workers weeks to build miles of a highwa, how long will it take 0 workers to build miles of highwa?. A Home Depot finds that the demand for a special power tool varies inversel with the advertised price of the tool. If the price is advertised at $8, there is a monthl demand for 0,000 units at all participating stores. Find the projected demand if the price is lowered to $0.8.. The electrical resistance of a copper wire varies directl with its length and inversel with the square of the diameter of the wire. If a wire 0 meters long with a diameter of millimeters has a resistance of ohms, find the resistance of a wire 0 meters long with a diameter of. millimeters 8. The intensit of light varies inversel as the square of the distance from the source. Suppose our reading light is placed feet from the surface of the book ou are reading. At what distance would the intensit of the light be twice as great?. Homework Answers:. (a) (b) (c) {, }. mph. 890 students k z kz kz. 08. feet. (a) kz (b) (c) (d).. w + w 8. a 9. C pounds. 8 feet.. seconds. ergs. of the original intensit. 0 weeks.,000 units.. ohms 8.. feet Page (Section.)
2.1 The Rectangular Coordinate System
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