# 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) = 0 (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 point-slope form to write an equation of a line use slope-intercept 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 Point-Slope Form The point-slope form of the equation of a nonvertical line with slope m that passes through, ) is ( = m( ) Slope-Intercept Form The slope-intercept 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 point-slope form. Then write the equation in slope-intercept 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 point-slope form and then in slope-intercept 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 Point-Slope Method Slope-Intercept 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 = 0. 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 slope-intercept 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) = 0 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 slope-intercept 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 slope-intercept 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) + 8 = 0 and = (c) + = and = (d) + = and =. Use the given conditions to write an equation for each line in slope-intercept 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 point-slope form. (b) Use this information to find the equation of the line that models this data in slope-intercept 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 ; = 0 ; (b) = + ; + = 0 ; (c) = + ; + = 0 = + ; + = 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 . Distance and Midpoint Formulas; Circles In this section ou will learn to: find the distance between two points find the midpoint of a line segment find the center and radius of a circle convert the general form of a circle s equation to standard form Distance Formula: The distance, d, between the points (, ) and (, ) in the rectangular coordinate sstem is d = ( ) + ( ) Recall: Pthagorean Theorem for right triangles: a + b = c Eample : Use the Pthagorean Theorem to find the distance from (, ) to (, ). 8 Eample : Find the distance between the following sets of points using the distance formula. (a) (-, -) and (, -) (b), and, Page (Section.)

14 Midpoint Formula: The coordinates of the midpoint of a line segment with endpoints + + (, ) and (, ) are:, Note: When finding the midpoint of a line segment, ou are finding the average of the - and -values. Eample : Find the midpoint of a line segment whose endpoints are, and,. Definition of a Circle: A circle is the set of all points in a plane that are equidistant (same distance) from a fied point, called the center. This fied distance from the center of the circle is called the radius. Standard Form of the Equation of a Circle: General From of the Equation of a Circle: ( h) + ( k) = r + + D + E + F = 0 (with center (h, k) and radius = r) (D, E, and F are real numbers) Eample : Find the equation of a circle with center (-, ) and radius =. Write the equation in standard and general form. Eample : Find the equation of the circle shown at the right. Write the equation in standard and general form. Write its domain (-values) and range (-values) using interval notation. 8 8 Page (Section.)

15 Eample : The endpoints of the diameter of a circle are (-, ) and (, -). Find the equation of the circle and write its domain (-values) and range (-values) using interval notation. 8 8 Recall Perfect Square Trinomials: + 9 = ( ) = ( + ) 8 + = ( ) + + = ( + ) Eample : Write the equation = 0 in standard form. Then find the center and radius of the circle. Also find the domain and range of the circle. Steps:. Move constant term to right side; group and terms on left side.. Form perfect square trinomials (completing the square).. Factor on left side; add on right side.. Find center (h, k) and radius. Page (Section.)

16 . Homework Problems. Find the distance between the points. (a) (, 9) and (9, ) (b) (-, ) and (, -) (c) (, ) and (, ) (d), and,. Find the midpoint of the line segment with the endpoints given below. (a) (, -) and (-8, -) (b) (-, -) and (, -8) (c) (, ) and (, ) (d), and,. Write the standard and the general form for the equation of the circle with the given center and radius. (a) (0, 0); r = (b) (-, ); r = (c) (-, 0); r =. The endpoints of the diameter of a circle are (-, ) and (, 0). (a) Find the coordinates of the circle s center. (b) Find the radius of the circle. (c) Write the equation of the circle in standard form. (d) Write the equation of the circle in general form. (e) Find the domain (-values) and range (-values) of the circle using interval notation.. Find the center, radius, domain, and range of each circle. (a) + = (b) ( + ) + = (c) ( + ) + ( ) = 9. Find the center and radius of each circle. (a) = 0 (b) + + = 0 (c) = 0 (d) + = 0. Homework Answers:. (a) ; (b) ; (c) 9 ; (d). (a), ; (b), ; (c) (, ) ; (d),. (a) + = ; + = 0 ; (b) ( + ) + ( ) = ; = 0 ; (c) ( + ) + = ; = 0. (a) (0, ); (b) ; (c) + ( ) = ; + + = 0; (e) D: [-, ]; R: [, ]. (a) (0, 0); r = 8; D: [-8, 8]; R: [-8, 8]; (b) (-, 0); r = ; D: 9, ; R:, ; (c) (-, ); r = ; D: [-, 0]; R: [-, 8]. (a) (, -); (b) (-, ); (c), ; (d), ; Page (Section.)

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