Appendix C: Review of Graphs, Equations, and Inequalities

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1 Appendi C: Review of Graphs, Equations, and Inequalities C. What ou should learn Just as ou can represent real numbers b points on a real number line, ou can represent ordered pairs of real numbers b points in a plane called the rectangular coordinate sstem, or the Cartesian plane, after the French mathematician René Descartes (9 0). The Cartesian plane is formed b using two real number lines intersecting at right angles, as shown in Figure C.. The horizontal real number line is usuall called the -ais, and the vertical real number line is usuall called the -ais. The point of intersection of these two aes is the origin, and the two aes divide the plane into four parts called quadrants. Quadrant II Origin (Vertical number line) Quadrant III The Distance Formula can be used to find lengths in real-life situations. For instance, in Eercise 89 on page C9, ou will use the Distance Formula to find the length of a football pass. Directed distance (, ) -ais Figure C. Quadrant I Plot points in the Cartesian plane. Use the Distance Formula to find the distance between two points. Use the Midpoint Formula to find the midpoint of a line segment. Find the equation of a circle. Translate points in the plane. Wh ou should learn it -ais -ais Directed distance (Horizontal number line) Quadrant IV -ais Ordered Pair (, ) Figure C. Each point in the plane corresponds to an ordered pair 共, 兲 of real numbers and, called coordinates of the point. The -coordinate represents the directed distance from the -ais to the point, and the -coordinate represents the directed distance from the -ais to the point, as shown in Figure C.. Directed distance from -ais 共, 兲 Directed distance from -ais The notation (, ) denotes both a point in the plane and an open interval on the real number line. The contet will tell ou which meaning is intended. Eample Plotting Points in the Cartesian Plane Plot the points 共, 兲, 共, 兲, 共0, 0兲, 共, 0兲, and 共, 兲 in the Cartesian plane. (, ) To plot the point 共, 兲, imagine a vertical line through on the -ais and a horizontal line through on the -ais. The intersection of these two lines is the point 共, 兲. This point is one unit to the left of the -ais and two units up from the -ais. The other four points can be plotted in a similar wa, as shown in Figure C.. Now tr Eercise. (, ) (0, 0) (, 0) (, ) Figure C. C

2 C Appendi C Review of Graphs, Equations, and Inequalities The Distance Formula Recall from the Pthagorean Theorem that, for a right triangle with hpotenuse of length c and sides of lengths a and b, ou have a b c Pthagorean Theorem as shown in Figure C.. (The converse is also true. That is, if a b c, then the triangle is a right triangle.) a + b = c a c Figure C. b Suppose ou want to determine the distance d between two points, and, in the plane. With these two points, a right triangle can be formed, as shown in Figure C.. (, ) d (, ) (, ) Figure C. The length of the vertical side of the triangle is Length of vertical side and the length of the horizontal side is. B the Pthagorean Theorem, Length of horizontal side d d d. This result is called the Distance Formula. The Distance Formula The distance d between the points, and, in the plane is d.

3 Appendi C. C Eample Finding a Distance Find the distance between the points 共, 兲 and 共, 兲. Algebraic Let 共, 兲 共, 兲 and 共, 兲 共, 兲. Then appl the Distance Formula as follows. d 冪共 兲 共 兲 Distance Formula 冪关 共 兲兴 共 兲 Substitute for,,, and. 冪共兲 共兲 Simplif. 冪 Simplif..8 Use a calculator. Graphical Use centimeter graph paper to plot the points A共, 兲 and B共, 兲. Carefull sketch the line segment from A to B. Then use a centimeter ruler to measure the length of the segment. So, the distance between the points is about.8 units. You can use the Pthagorean Theorem to check that the distance is correct.? d Pthagorean Theorem? Substitute for d. 共冪 兲 Distance checks. Cm Figure C. The line segment measures about.8 centimeters, as shown in Figure C.. So, the distance between the points is about.8 units. Now tr Eercise. When the Distance Formula is used, it does not matter which point is 共, 兲 and which is 共, 兲, because the result will be the same. For instance, in Eample, let 共, 兲 共, 兲 and 共, 兲 共, 兲. Then d 冪共 兲 共 兲 冪共 兲 共 兲 冪.8. Eample Verifing a Right Triangle Show that the points 共, 兲, 共, 0兲, and 共, 7兲 are the vertices of a right triangle. (, 7) 7 d = The three points are plotted in Figure C.7. Using the Distance Formula, ou can find the lengths of the three sides as follows. d 冪共 兲 共7 兲 冪9 冪 d 冪共 兲 共0 兲 冪 冪 d 冪共 兲 共7 0兲 冪 9 冪0 d = 共d兲 共d兲 0 共d兲 ou can conclude that the triangle must be a right triangle. Now tr Eercise. 0 d = (, ) (, 0) Figure C.7 Because 7

4 C Appendi C Review of Graphs, Equations, and Inequalities The Midpoint Formula To find the midpoint of the line segment that joins two points in a coordinate plane, find the average values of the respective coordinates of the two endpoints using the Midpoint Formula. The Midpoint Formula The midpoint of the line segment joining the points 共, 兲 and 共, 兲 is given b the Midpoint Formula Midpoint 冢,. 冣 Eample Finding a Line Segment s Midpoint Find the midpoint of the line segment joining the points 共, 兲 and 共9, 兲. Let 共, 兲 共, 兲 and 共, 兲 共9, 兲. Midpoint 冢, 冢 9, 冣 (9, ) (, 0) Midpoint Formula 冣 共, 0兲 (, ) Substitute for,,, and. Figure C.8 Now tr Eercise 7. Eample Estimating Annual Revenues Verizon Communications had annual revenues of $88. billion in 00 and $97. billion in 008. Without knowing an additional information, what would ou estimate the 007 revenue to have been? (Source: Verizon Communications) 冢00 008, 冣 So, ou would estimate the 007 revenue to have been about $9.8 billion, as shown in Figure C.9. (The actual 007 revenue was $9. billion.) Now tr Eercise. Verizon Communications Annual Revenues Revenues (in billions of dollars) One solution to the problem is to assume that revenue followed a linear pattern. With this assumption, ou can estimate the 007 revenue b finding the midpoint of the line segment connecting the points 共00, 88.兲 and 共008, 97.兲. 共007, 9.8兲 Simplif. The midpoint of the line segment is 共, 0兲, as shown in Figure C.8. Midpoint (008, 97.) (007, 9.8) (00, 88.) Year Figure C.9 Midpoint 008 Midpoint 9

5 Appendi C. C The Equation of a Circle The Distance Formula provides a convenient wa to define circles. A circle of radius r with center at the point 共h, k兲 is shown in Figure C.0. The point 共, 兲 is on this circle if and onl if its distance from the center 共h, k兲 is r. This means that a circle in the plane consists of all points 共, 兲 that are a given positive distance r from a fied point 共h, k兲. Using the Distance Formula, ou can epress this relationship b saing that the point 共, 兲 lies on the circle if and onl if 冪共 h兲 共 k兲 r. B squaring each side of this equation, ou obtain the standard form of the equation of a circle. Center: (h, k) Radius: r Point on circle: (, ) Figure C.0 Standard Form of the Equation of a Circle The standard form of the equation of a circle is 共 h兲 共 k兲 r. The point 共h, k兲 is the center of the circle, and the positive number r is the radius of the circle. The standard form of the equation of a circle whose center is the origin, 共h, k兲 共0, 0兲, is r. Eample 7 Writing an Equation of a Circle The point 共, 兲 lies on a circle whose center is at 共, 兲, as shown in Figure C.. Write the standard form of the equation of this circle. 8 The radius r of the circle is the distance between 共, 兲 and 共, 兲. r 冪关 共 兲兴 共 兲 Substitute for,, h, and k. 冪 Simplif. 冪 Simplif. 冪0 Radius (, ) Equation of circle 关 共 兲兴 共 兲 共冪0 兲 共 兲 共 兲 0. Now tr Eercise 7. Substitute for h, k, and r. Standard form Using 共h, k兲 共, 兲 and r 冪0, the equation of the circle is 共 h兲 共 k兲 r (, ) Figure C.

6 C Appendi C Review of Graphs, Equations, and Inequalities Application Much of computer graphics consists of transformations of points in a coordinate plane. One tpe of transformation, a translation, is illustrated in Eample 8. Other tpes of transformations include reflections, rotations, and stretches. Eample 8 Translating Points in the Plane The triangle in Figure C. has vertices at the points 共, 兲, 共, 兲, and 共, 兲. Shift the triangle three units to the right and two units upward and find the vertices of the shifted triangle, as shown in Figure C.. (, ) (, ) 7 7 (, ) Figure C. Figure C. To shift the vertices three units to the right, add to each of the -coordinates. To shift the vertices two units upward, add to each of the -coordinates. Original Point 共, 兲 Translated Point 共, 兲 共, 兲 共, 兲 共, 兲 共, 兲 共, 兲 共, 兲 共, 兲 Plotting the translated points and sketching the line segments between them produces the shifted triangle shown in Figure C.. Now tr Eercise 8. Eample 8 shows how to translate points in a coordinate plane. The following transformed points are related to the original points as follows. Original Point Transformed Point 共, 兲 共, 兲 共, 兲 is a reflection of the original point in the -ais. 共, 兲 共, 兲 共, 兲 is a reflection of the original point in the -ais. 共, 兲 共, 兲 共, 兲 is a reflection of the original point through the origin. The figures provided with Eample 8 were not reall essential to the solution. Nevertheless, it is strongl recommended that ou develop the habit of including sketches with our solutions because the serve as useful problem-solving tools.

7 Appendi C. C7 C. Eercises For instructions on how to use a graphing utilit, see Appendi A. Vocabular and Concept Check In Eercises, fill in the blank(s).. An ordered pair of real numbers can be represented in a plane called the rectangular coordinate sstem or the plane.. The is a result derived from the Pthagorean Theorem.. Finding the average values of the respective coordinates of the two endpoints of a line segment in a coordinate plane is also known as using the.. The standard form of the equation of a circle is, where the point h, k is the of the circle and the positive number r is the of the circle. In Eercises 0, match each term with its definition.. -ais (a) point of intersection of vertical ais and horizontal ais. -ais (b) directed distance from the -ais 7. origin (c) horizontal real number line 8. quadrants (d) four regions of the coordinate plane 9. -coordinate (e) directed distance from the -ais 0. -coordinate (f) vertical real number line Procedures and Problem Solving Approimating Coordinates of Points In Eercises and, approimate the coordinates of the points... A C D B C Plotting Points in the Cartesian Plane In Eercises, plot the points in the Cartesian plane..,,,, 0,,,.,, 0, 0,, 0,,., 8, 0.,,,,,..,,,,,,, Finding Coordinates of Points In Eercises 7 0, find the coordinates of the point. 7. The point is located five units to the left of the -ais and four units above the -ais. 8. The point is located three units below the -ais and two units to the right of the -ais. 9. The point is on the -ais and si units below the -ais. 0. The point is on the -ais and units to the left of the -ais. D B A Determining Quadrants In Eercises 0, determine the quadrant(s) in which, is located so that the condition(s) is (are) satisfied.. > 0 and < 0. < 0 and < 0. and > 0. > and. <. > 7. < 0 and > 0 8. > 0 and < 0 9. > 0 0. < 0 Finding a Distance In Eercises 0, find the distance between the points algebraicall and confirm graphicall b using centimeter graph paper and a centimeter ruler..,,,., 0,, 7.,,,.,,,.,,,. 8,, 0, 0 7.,,, 8.,,, 9..,.,., ,.,.9, 8. Verifing a Right Triangle In Eercises, (a) find the length of each side of the right triangle and (b) show that these lengths satisf the Pthagorean Theorem... (, ) (, ) (, ) 8 (, 0) (, ) 8 (, 0)

8 C8 Appendi C Review of Graphs, Equations, and Inequalities.. (9, ) (9, ) (, ) 8 Verifing a Polgon In Eercises, show that the points form the vertices of the polgon.. Right triangle:, 0,,,,. Right triangle:,,,,, 7. Isosceles triangle:,,,,, 8. Isosceles triangle:,,, 9,, 7 9. Parallelogram:,, 0, 9,, 0, 0, 0. Parallelogram: 0,,, 7,,,,. Rectangle:,, 0, 8,,,, (Hint: Show that the diagonals are of equal length.). Rectangle:,,,,,,, (Hint: Show that the diagonals are of equal length.) Finding a Line Segment s Midpoint In Eercises, (a) plot the points, (b) find the distance between the points, and (c) find the midpoint of the line segment joining the points.. 0, 0, 8,.,,, 0., 0,,. 7,,, 8 7.,,, 8., 0, 0, 9.,,, 0.,,,..,.,.7,.8..8,.,.,.9 Estimating Annual Revenues In Eercises and, use the Midpoint Formula to estimate the annual revenues (in millions of dollars) for Teas Roadhouse and Papa John s in 00. The revenues for the two companies in 00 and 009 are shown in the tables. Assume that the revenues followed a linear pattern. (Sources: Teas Roadhouse, Inc.; Papa John s International). Teas Roadhouse. Papa John s Intl. (, ) (, ) (, ) Year Annual revenue (in millions of dollars) Year Annual revenue (in millions of dollars) Writing an Equation of a Circle In Eercises 78, write the standard form of the equation of the specified circle.. Center: 0, 0 ; radius:. Center: 0, 0 ; radius: 7. Center:, ; radius: 8. Center:, ; radius: 9. Center:, ; solution point: 0, Center:, ; solution point:, 7. Endpoints of a diameter: 0, 0,, 8 7. Endpoints of a diameter:,,, 7. Center:, ; tangent to the -ais 7. Center:, ; tangent to the -ais 7. The circle inscribed in the square with vertices 7,,,,, 0, and 7, 0 7. The circle inscribed in the square with vertices, 0, 8, 0, 8, 0, and, Sketching a Circle In Eercises 79 8, find the center and radius, and sketch the circle Translating Points in the Plane In Eercises 8 88, the polgon is shifted to a new position in the plane. Find the coordinates of the vertices of the polgon in the new position (, ) 7 (, ) units (, ) (, ) units units (, ) units 7 (, 0) (, ) 87. Original coordinates of vertices: 0,,,,,,, Shift: three units upward, one unit to the left 88. Original coordinates of vertices:,,,,, Shift: two units downward, three units to the left

9 Appendi C. C9 89. (p. C) In a football game, a quarterback throws a pass from the -ard line, 0 ards from the sideline, as shown in the figure. The pass is caught on the 0-ard line, ards from the same sideline. How long is the pass? 90. Phsical Education A major league baseball diamond is a square with 90-foot sides. In the figure, home plate is at the origin and the first base line lies on the positive -ais. The right fielder fields the ball at the point 00,. How far does the right fielder have to throw the ball to get a runner out at home plate? How far does the right fielder have to throw the ball to get a runner out at third base? (Round our answers to one decimal place.) Distance (in feet) Distance (in ards) 9. Aviation A jet plane flies from Naples, Ital in a straight line to Rome, Ital, which is 0 kilometers north and 0 kilometers west of Naples. How far does the plane fl? Conclusions (0, 90) (0, 0) (0, ) (, 0) Distance (in ards) (00, ) Distance (in feet) True or False? In Eercises 9 9, determine whether the statement is true or false. Justif our answer. 9. The points 8,,,, and, represent the vertices of an isosceles triangle. 9. If four points represent the vertices of a polgon, and the four sides are equal, then the polgon must be a square. 9. In order to divide a line segment into equal parts, ou would have to use the Midpoint Formula times. 9. Think About It When plotting points on the rectangular coordinate sstem, is it true that the scales on the - and -aes must be the same? Eplain. 97. Eploration A line segment has, as one endpoint and m, m as its midpoint. Find the other endpoint, of the line segment in terms of,, m, and m. Use the result to find the coordinates of the endpoint of a line segment when the coordinates of the other endpoint and midpoint are, respectivel, (a),,,. (b),,,. 98. Eploration Use the Midpoint Formula three times to find the three points that divide the line segment joining, and, into four parts. Use the result to find the points that divide the line segment joining the given points into four equal parts. (a),,, (b),, 0, Proof Prove that the diagonals of the parallelogram in the figure intersect at their midpoints. (b, c) (0, 0) (a + b, c) (a, 0) 00. CAPSTONE Use the plot of the point 0, 0 in the figure. Match the transformation of the point with the correct plot. Eplain our reasoning. [The plots are labeled (i), (ii), (iii), and (iv).] (i) (iii) ( 0, 0 ) (ii) (iv) 9. Think About It What is the -coordinate of an point on the -ais? What is the -coordinate of an point on the -ais? (a) (c) 0, 0 0, 0 (b) 0, 0 (d) 0, 0

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