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Coordinate Geometry and Functions The principal goal of education is to create individuals who are capable of doing new things, not simply of repeating what other generations have done. Jean Piaget Swiss psychologist Introduction to the Cartesian Coordinate (x,y) Plane Before the end of the European Renaissance, math was cleanly divided into the two separate subjects of geometry and algebra. You didn't use algebraic equations in geometry, and you didn't draw any pictures in algebra. Then, around 1637, a French guy named René Descartes came up with a way to put these two subjects together Seating Chart The following seating chart has 5 rows of desks with 4 desks in each row Mary sits in the third row at the second desk Row 1 Row 2 Row 3 Row 4 Row 5 Who is sitting in desk (4,2)? N 4 A B C D E 3 F G H I J Row 4 Seat 2 2 K L M N O 1 P Q R S T 1 2 3 4 5 Mary sits in the third row at the second desk (3,2) and George sits in the second row at the third desk (2,3). Are these seats the same? No!! The seats (3,2) and (2,3) are called ordered pairs because the order in which the pair of numbers is written is important!! Coordinate Plane This was a handy way to reference individuals in the room, telling you how far over (row 2) and how far up (seat 4) you needed to go. Descartes did something similar for mathematics Coordinate Plane Descartes' breakthrough was in taking a second number line, standing it up on its end, and crossing the first number line at zero: Definition x axis – a horizontal number line on a coordinate grid. 0 1 2 3 4 5 6 x Definition y axis – a vertical number line on a coordinate grid. y 6 5 4 3 2 1 0 Coordinates – an ordered pair of numbers that give the location of a point on a grid. (3, 4) 6 5 (3,4) 4 3 2 1 00 1 2 3 4 5 6 Hint The first number is always the x or first letter in the alphabet. The second number is always the y the second letter in the alphabet. 6 5 (3,4) 4 3 2 1 00 1 2 3 4 5 6 How to Plot Ordered Pairs Step 1 – Always find the x value first, moving horizontally either right (positive) or left (negative). y 6 (2, 3) 5 4 3 2 1 00 1 2 3 4 5 6 x How to Plot Ordered Pairs Step 2 – Starting from your new position find the y value by moving vertically either up (positive) or down (negative). y 6 (2, 3) 5 4 (2,3) 3 2 1 00 1 2 3 4 5 6 x What is the ordered pair? (3,5) y 6 5 4 3 2 1 00 1 2 3 4 5 6 x The x-axis and y-axis separate the coordinate plane into four regions, called quadrants. II (-, +) I (+, +) III (-, -) IV (+, -) How to Plot in 4 Quadrants Step 1 - Plot the x number first moving to the left when the number is negative. (-3, -2) y 3 2 1 x 0 -1 -2 -3 -3 -2 -1 0 1 2 3 How to Plot in 4 Quadrants Step 2 - Plot the y number moving from your new position down 2 when the number is negative. y 3 (-3, -2) 2 1 0 x -1 -2 -3 -3 -2 -1 0 1 2 3 Write the ordered pairs that name points A, B, C, and D. 5 A = (1, 3) B = (3, -2) C = (0, -4) D = (-6, -1) •A 5 -5 •D •B -5 •C Linear Equation: an equation whose graph forms a line in an x,y coordinate plane In linear equations, all variables are taken to the first power. x+y=1 3x – 4y = 2 Suppose you were asked to graph the solution set of the linear equation, x+y=3 You could start by creating a table of values. Choose any number to use as x or y x+y=3 x y 0 3 3 0 -3 6 Substitute into equation and solve -3 0+y =3 3 y ==0 36 Plot the points and graph x+y=3 x y 0 3 3 0 -3 6 (-3,6) (0,3) connect the points with a straight line (3,0) Let’s graph another line: 3x – y = 2 x y -1 -5 0 -2 2 4 Substitute into equation and solve 3(-1)––yy== 2 3(0) 3(2) y=4 -5 -2 Plot the points and graph 3x – y = 2 (2,4) x y -1 -5 0 -2 2 4 connect the points with a straight line (0,-2) (-1,-5) Graphing Using Intercepts The x-intercept of a line is the x-coordinate of the point where the line intercepts the x-axis. 8 The line shown intercepts the x-axis at (2,0). 6 4 2 We say that the x-intercept is 2. -8 -6 -4 -2 -2 -4 -6 -8 2 4 6 8 Graphing Using Intercepts The y-intercept of a line is the y-coordinate of the point where the line intercepts the y-axis. The line shown intercepts the y-axis at (0,-6). 8 6 4 We say that the y-intercept is -6. 2 -8 -6 -4 -2 -2 -4 -6 -8 2 4 6 8 Graph using intercepts. 4x – 3y = 12 *To find the y-intercept, let x = 0. 4x – 3y = 12 4(0) – 3y = 12 0 – 3y = 12 -3y = 12 -3 -3 y = -4 x 0 y -4 Graph using intercepts. 4x – 3y = 12 *To find the x-intercept, let y = 0. 4x – 3y = 12 4x – 3(0) = 12 4x - 0 = 12 4x = 12 4 4 x=3 x 0 3 y -4 0 Graph using intercepts. 4x – 3y = 12 8 x 0 3 6 4 2 -8 -6 -4 -2 -2 -4 -6 -8 2 4 6 8 y -4 0 Slope Graphs of any line can easily be classified by its y-intercept and by its steepness. The steepness is often referred to as slope Types of Slope Zero Negative Positive Undefined or No Slope Slope 1 1 To walk up the steps, the man must go up one step and over one step 1 To walk up the steps, the man must go up two steps and over one 1 step To walk up the steps, the man must go up one step and over two steps Slope 10 rise Slope = run run 8 6 4 rise 2 -4 -2 -2 -4 -6 -8 -10 2 4 6 8 10 Graph the line slope containing points (2,1) and (7,6) Find the and find the slope. rise 5 Slope = 1 run 5 10 8 run 6 rise 4 2 -4 -2 -2 -4 -6 -8 -10 2 4 6 8 10 Slope rise change in y coordinates Slope = change in x coordinates run y2 y1 m x2 x1 or y1 y2 m x1 x2 Find the slope of the line containing points (1,6) and (5,4). y2 y1 m x2 x1 46 m 5 1 2 m 4 1 m 2 Slope of a Horizontal Line 10 What is the rise? What is the run? 0 8 6 rise 0 Slope = 0 run 6 The slope of ANY horizontal line is 0. 6 4 2 -4 -2 -2 -4 -6 -8 -10 2 4 6 8 10 Slope of a Vertical Line 10 What is the rise? What is the run? 5 8 6 0 rise 5 Slope = run 0 The slope of ANY vertical line is undefined. 4 2 -4 -2 -2 -4 -6 -8 -10 2 4 6 8 10 If given an equation of a line, there is a nice way to find the slope and yintercept. Simply write the equation in slope-intercept form, which is: y = mx + b slope y-intercept Find the slope and y-intercept of the following equations. y = 3x + ½ slope= 3 y-intercept = ½ Find the slope and yintercept: 3x + 5y = 10 First, solve the equation for y. 5y = -3x + 10 y = -3/5 x + 2 m= -3/5 b=2 Find the equation of a line: To find the equation, consider slope intercept form: y = mx + b • • • • • • Slope If given, just substitute it into the equation Otherwise calculate the slope y-intercept If given, just substitute it into the equation Otherwise use a point and substitute into equation for x and for y, then solve for b Find the equation of a line containing the following points: (4,2) and (3,6) m=? b=? Always find slope first! Find the slope using the formula. m=2–6 4–3 m = -4 y = -4x + b (4, 2) and (3, 6) m = -4 Find the y-intercept. y = mx + b 2 = -4 ͯ 4 + b 2 = -16 + b 18 = b Pick a point to plug into the equation (4,2) → x=4 y=2 Write equation: y = -4x + 18 Relations A relation is a mapping, or pairing, of input values with output values. The set of input values is called the domain (x values) The set of output values is called the range (y-values) Functions A relation is a function provided there is exactly one output for each input. It is NOT a function if at least one input has more than one output In order for a relationship to be a function… EVERY INPUT MUST HAVE AN OUTPUT TWO DIFFERENT INPUTS CAN HAVE THE SAME OUTPUT INPUT (DOMAIN) FUNCTION MACHINE ONE INPUT MUST HAVE ONLY ONE OUTPUT OUTPUT (RANGE) Which of the following relations are functions? R= {(9,10, (-5, -2), (2, -1), (3, -9)} S= {(6, a), (8, f), (6, b), (-2, p)} T= {(z, 7), (y, -5), (r, 7) (z, 0), (k, 0)} No two ordered pairs can have the same first coordinate (and different second coordinates). Identify the Domain and Range. Then tell if the relation is a function. Input Output -3 3 1 1 3 -2 4 Domain = {-3, 1,3,4} Range = {3,1,-2} Function? Yes: each input is mapped onto exactly one output Identify the Domain and Range. Then tell if the relation is a function. Input Output -3 3 1 -2 4 1 4 Domain = {-3, 1,4} Range = {3,-2,1,4} Function? No: input 1 is mapped onto both -2 & 1 The Vertical Line Test If it is possible for a vertical line to intersect a graph at more than one point, then the graph is NOT the graph of a function. Page 117 Which are functions? Function Not a Function Function Function Notation f (x) “f of x” Input = x Output = f(x) = y Before… Now… y = 6 – 3x f(x) = 6 – 3x x y x f(x) -2 12 -2 12 -1 9 -1 9 0 6 0 6 1 3 1 3 2 0 2 0 (x, y) (x, f(x)) Evaluate the function: f(x) = 2x2 – 3 Find f(0), f(3) f(0) = 2(0)2 – 3 f(3) = 2(3)2 – 3 =0–3 = 18 – 3 =-3 = 15