Graphing Equations. The Rectangular Coordinate System

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1 3.1 Graphing Equations The Rectangular Coordinate Sstem Ordered pair two numbers associated with a point on a graph. The first number gives the horizontal location of the point. The second gives the vertical location. Coordinate a number in an ordered pair; - coordinate, -coordinate. -ais horizontal number line -ais vertical number line Origin point of intersection of the two aes Quadrants four regions created b the intersection of the two aes. 1

2 Graphing an Ordered Pair To graph the point corresponding to a particular ordered pair (a, b), ou must start at the origin and move a units to the left or right (right if a is positive, left if a is negative), then move b units up or down (up if b is positive, down if b is negative). Graphing an Ordered Pair Quadrant II ( 4, 2) ( 6, 0) -ais Quadrant I (0, 5) (5, 3) 3 units up (0, 0) 5 units right -ais origin (2, 4) Quadrant III Quadrant IV Note that the order of the coordinates is ver important, since ( 4, 2) and (2, 4) are located in different positions. 2

3 Plot each ordered pair. State in which quadrant, or on which ais the point lies. a. (4, 2) b. ( 3, 2) c. (2, 3) d. (0, 4) e. (5, 0) Solution a. (4, 2) Quadrant I b. ( 3, 2) Quadrant III c. (2, 3) Quadrant IV d. (0, 4) -ais e. (5, 0) -ais Vocabular Paired data are data that can be represented as ordered pairs. A scatter diagram is the graph of paired data as points in the rectangular coordinate sstem. 3

4 Completing Ordered Pair Solutions In general, an ordered pair is a solution of an equation in two variables if replacing the variables b the values of the ordered pair results in a true statement. If ou know one coordinate of an ordered pair that is a solution for an equation, ou can find the other coordinate through substitution and solving the resulting equation. Determine whether (3, 2) is a solution of = 4. Let = 3 and = 2 in the equation = 4 2(3) + 5( 2) = ( 10) = 4 4 = 4 True So (3, 2) is a solution of = 4 4

5 Determine whether ( 1, 6) is a solution of 3 = 5. Let = 1 and = 6 in the equation. 3 = 5 3( 1) 6 = = 5 9 = 5 False So ( 1, 6) is not a solution of 3 = 5 Linear Equations A linear equation in two variables is an equation that can be written in the form A + B = C where A, B, and C are real numbers and A and B not both 0. This form is called standard form. 5

6 Graph the linear equation 2 = 4. To graph this equation, we find three ordered pair solutions b choosing a value for one of the variables, or, then solving for the other variable. (The third solution acts as a check for the other two.) We plot the solution points, then draw the line containing the 3 points. (cont) Graph the linear equation 2 = 4. Let = 1. 2 = 4 2(1) = 4 Replace with 1. 2 = 4 Simplif. = 6 Subtract 2 from both sides. = 6 Multipl both sides b 1. The ordered pair (1, 6) is a solution of 2 = 4. 6

7 (cont) Graph the linear equation 2 = 4. Net, let = 4. 2 = = 4 Replace with 4. 2 = Add 4 to both sides. 2 = 0 Simplif. = 0 Divide both sides b 2. The ordered pair (0, 4) is a second solution. Graph the linear equation 2 = 4. Net, let = 4. (cont) 2 = 4 2( 3) = 4 Replace with 3. 6 = 4 Simplif. = 2 Add 6 to both sides. = 2 Multipl both sides b 1. The third solution is ( 3, 2). 7

8 (cont) (0, 4) (1, 6) Now we plot all three of the solutions (1, 6), (0, 4) and ( 3, 2). And then we draw the line that contains the three points. ( 3, 2) Graph the linear equation Since the equation is solved for, we should choose values for. To avoid fractions, we should select values of that are multiples of 4 (the denominator of the fraction). 8

9 (cont) Graph the linear equation Let = 4. = = (4) + 3 Replace with 4. 4 = = 6 Simplif. So one solution is (4, 6). (cont) 3 Graph the linear equation 3 4 Net, let = 0. 3 = = 3 (0) + 3 Replace with 0. 4 = = 3 Simplif. So a second solution is (0, 3). 9

10 (cont) 3 Graph the linear equation 3 4 Net, let = 4. 3 = = ( 4) + 3 Replace with 4. 4 = = 0 Simplif. So the third solution is ( 4, 0). (cont) Now we plot all three of the ordered pair solutions; (4, 6), (0, 3) and ( 4, 0). And then we draw the line that contains the three points. (0, 3) ( 4, 0) (4, 6) 10

11 Intercepts Finding - and -Intercepts To find the -intercept, let = 0 and solve for. To find the -intercept, let = 0 and solve for. Graph the linear equation 1 3 Let = 0 Let = 6 1 ( ) 1 ( ) Let = 3 1 ( )

12 Graphing Nonlinear Equations Not all equations in two variables are linear equations, and not all graphs of equations in two variables are lines. Graph the parabola 2. 5 =

13 Graph the absolute value equation. 5 = Graph the equation

14 3.2 Introduction to Functions Vocabular A set of ordered pairs is also called a relation. The domain is the set of -coordinates (first components) of the ordered pairs. The range is the set of -coordinates (second components) of the ordered pairs. 14

15 Determine the domain and range of the relation {(4,9), ( 4,9), (2,3), (10, 5)} Domain is the set of all the first coordinates of the ordered pairs: {4, 4, 2, 10} Range is the set of all the second coordinates of the ordered pairs: {9, 3, 5} Find the domain and range of the following relation. Input (Animal) Polar Bear Cow Chimpanzee Giraffe Gorilla Kangaroo Red Fo Output (Life Span)

16 (cont) Domain is {Polar Bear, Cow, Chimpanzee, Giraffe, Gorilla, Kangaroo, Red Fo} Range is {20, 15, 10, 7} Functions Some relations are also functions. A function is a relation in which each first component in the ordered pairs corresponds to eactl one second component. 16

17 Is the relation{(4,9), ( 4,9), (2,3), (10, 5)}, also a function? Since each element of the domain is paired with onl one element of the range, it is a function. Note: It s oka for a -value to be assigned to more than one -value, but an -value cannot be assigned to more than one -value (has to be assigned to ONLY one -value). Is the relation = 2 2 a function? Since each element of the domain (the -values) would produce onl one element of the range (the -values), it is a function. 17

18 Is the relation 2 2 = 9 a function? Since each element of the domain (the -values) would correspond with 2 different values of the range (both a positive and negative -value), the relation is NOT a function. Vertical Line Test Graphs can be used to determine if a relation is a function. Vertical Line Test If no vertical line can be drawn so that it intersects a graph more than once, the graph is not the graph of a function. 18

19 Use the vertical line test to determine whether the graph to the right is the graph of a function. Since no vertical line will intersect this graph more than once, it is the graph of a function. Use the vertical line test to determine whether the graph to the right is the graph of a function. Since no vertical line will intersect this graph more than once, it is the graph of a function. 19

20 Use the vertical line test to determine whether the graph to the right is the graph of a function. Since vertical lines can be drawn that intersect the graph in two points, it is NOT the graph of a function. Vertical Line Test Since the graph of a linear equation is a line, all linear equations are functions, ecept those whose graph is a vertical line Note: An equation of the form = c is a horizontal line and IS a function. An equation of the form = c is a vertical line and IS NOT a function. 20

21 Find the domain and range of the function graphed to the right. Use interval notation. Domain is [ 3, 4] Range is [ 4, 2] Range Domain Find the domain and range of the function graphed to the right. Use interval notation. Domain is (, ) Range is [ 2, ) Range Domain 21

22 Helpful Hint Note that f() is a special smbol in mathematics used to denote a function. The smbol f() is read f of. It does not mean f (f times ). 22

23 Find each function value. a. If f() = 4 1, find f(2). f(2) = 4(2) 1 = 7 b. If g() = 2 2, find g( 3). g( 3) = ( 3) 2 2( 3) = 9 ( 6) = 15 Given the graph of the following function, find each function value b inspecting the graph. f(5) = 8 f(4) = 3 f( 5) = 1 f( 6) = 7 f() 23

Graphs, Linear Equations, and Functions

Graphs, Linear Equations, and Functions. The Rectangular R. Coordinate Fractions Sstem bjectives. Interpret a line graph.. Plot ordered pairs.. Find ordered pairs that satisf a given equation. 4. Graph