Patterns, Equations, And Graphs

Patterns, Equations, and GraphsSection 1-9

GoalsGoal To use tables, equations,and graphs to describerelationships.

Vocabulary Solution of an equation Inductive reasoning

Review: Graphing in theCoordinate PlaneThe coordinate plane is formed bythe intersection of twoperpendicular number lines calledaxes. The point of intersection,called the origin, is at 0 on eachnumber line. The horizontalnumber line is called the x-axis,and the vertical number line iscalled the y-axis.

Graphing in the CoordinatePlanePoints on the coordinate plane are described using orderedpairs. An ordered pair consists of an x-coordinate and ay-coordinate and is written (x, y). Points are often namedby a capital letter.Reading MathThe x-coordinate tells how many units to move left or right fromthe origin. The y-coordinate tells how many units to move up ordown.

Example: Graphing in theCoordinate PlaneGraph each point.A. T(–4, 4)Start at the origin.Move 4 units left and 4units up.T(–4, 4) B. U(0, –5)Start at the origin.Move 5 units down.C. V (–2, –3)Start at the origin.Move 2 units left and 3 unitsdown. V(–2, 3) U(0, –5)

Your Turn:Graph each point.A. R(2, –3)Start at the origin.Move 2 units right and 3units down.T(–2,6)S(0,2)B. S(0, 2)Start at the origin.Move 2 units up.C. T(–2, 6)Start at the origin.Moveunits left and6 units2up. R(2, –3)

Graphing in the CoordinatePlaneThe axes divide thecoordinate plane intofour quadrants. Pointsthat lie on an axis arenot in any quadrant.

Example: Locating PointsName the quadrant in which each point lies.yA. EQuadrant llB. Fno quadrant (y-axis) F E GxC. GQuadrant lD. HQuadrant lll H

Your Turn:Name the quadrant in which each point lies.yA. T TB. UQuadrant lxC. VQuadrant lll VD. WQuadrant ll U Wno quadrant (y-axis)

The Rectangular Coordinate SystemSUMMARY: The Rectangular Coordinate System Composed of two real number lines – one horizontal (the x-axis) andone vertical (the y-axis). The x- and y-axes intersect at the origin. Also called the Cartesian plane or xy-plane. Points in the rectangular coordinate system are denoted (x, y) and arecalled the coordinates of the point. We call the x the x-coordinate andthe y the y-coordinate. If both x and y are positive, the point lies in quadrant I; if x isnegative, but y is positive, the point lies in quadrant II; if x isnegative and y is negative, the point lies in quadrant III; if x ispositive and y is negative, the point lies in quadrant IV. Points on the x-axis have a y-coordinate of 0; points on the y-axishave an x-coordinate of 0.

Equation in Two VariablesAn equation in two variables, x and y, is a statement in which thealgebraic expressions involving x and y are equal. The expressionsare called sides of the equation.x y 15x2 – 2y2 4y 1 4xAny values of the variables that make the equation a true statementare said to be solutions of the equation.x y 15The ordered pair (5, 10) is a solution of the equation.5 10 1515 15

Solutions to EquationsExample:Determine if the following ordered pairs satisfy theequation 2x y 5.a.) (2, 1)b.) (3, – 4)2x y 52x y 52(2) (1) 52(3) (– 4) 54 1 56 (– 4) 55 5True(2, 1) is a solution.2 5False(3, – 4) is not a solution.

Equation in Two VariablesAn equation that contains two variables can be used asa rule to generate ordered pairs. When you substitute avalue for x, you generate a value for y. The valuesubstituted for x is called the input, and the valuegenerated for y is called the output.OutputInputy 10x 5

Table of ValuesUse the equation y 6x 5 to complete the table and listthe ordered pairs that are solutions to the equation.x–202y(x, y)(– 2, – 7)(0, 5)(2, 17)x 0x 2y 6x 5y 6x 5y 6x 5y 6(– 2) 5y 6(0) 5y 6(2) 5y – 12 5y 0 5y 12 5y –7y 5y 17x –2

Example: ApplicationAn engraver charges a setup fee of 10 plus 2 for everyword engraved. Write a rule for the engraver’s fee. Writeordered pairs for the engraver’s fee when there are 5, 10,15, and 20 words engraved.Let y represent the engraver’s fee and x represent thenumber of words engraved.Engraver’s feeis 10plus 2for eachwordy 10 2·xy 10 2x

Writing MathThe engraver’s fee is determined by the numberof words in the engraving. So the number ofwords is the input and the engraver’s fee is theoutput.

Example: SolutionNumber ofWordsEngravedRuleChargesOrderedPairx (input)y 10 2xy (output)(x, y)5y 10 2(5)20(5, 20)10y 10 2(10)30(10, 30)15y 10 2(15)40(15, 40)20y 10 2(20)50(20, 50)

Your Turn:What if ? The caricature artist increased his fees. He nowcharges a 10 set up fee plus 20 for each person in thepicture. Write a rule for the artist’s new fee. Find theartist’s fee when there are 1, 2, 3 and 4 people in the picture.Let y represent the artist’s fee and x represent the number ofpeople in the picture.Artist’s feeyis 10 10y 10 20xplus 20 for each 20·personx

Solution:Number ofPeople inPictureRuleChargesOrderedPairx (input)y 10 20xy (output)(x, y)1y 10 20(1)30(1, 30)2y 10 20(2)50(2, 50)3y 10 20(3)70(3, 70)4y 10 20(4)90(4, 90)

Graphing Ordered PairsWhen you graph ordered pairs generated bya function, they may create a pattern.

Example: Graphing Ordered PairsGenerate ordered pairs for the function usingthe given values for x. Graph the ordered pairsand describe the pattern.y 2x 1; x –2, –1, 0, 1, 2Inputx–2–1012Outputy2(–2) 1 –32(–1) 1 –12(0) 1 12(1) 1 32(2) 1 5OrderedPair (x, y)(–2, –3)(–1, –1)(0, 1)(1, 3)(2, 5) The points form a line.

Your Turn: Generate ordered pairs for thefunction using the given values for x. Graph theordered pairs and describe the pattern.y 1 x – 4; x –4, –2, 0, 2, 42InputOutputOrderedPairxy(x, y)–2 – 4 –6–1 – 4 –50 – 4 –41 – 4 –3(–4, –6)(–2, –5)(0, –4)(2, –3)(4, –2)–4–20242 – 4 –2The points form a line.

Definition Inductive Reasoning – is the process ofreaching a conclusion based on an observedpattern.– Can be used to predict values based on apattern.

Inductive Reasoning Moves from specific observations to broadergeneralizations or predictions from a pattern.Steps:1.2.3.Observing data.Detect and recognizing patterns.Make generalizations or predictions from those patterns.PredictPatternObservation

Example: Inductive ReasoningMake a prediction about the next number based on thepattern.2, 4, 12, 48, 240Find a pattern:24 212 348 4240 5The numbers are multiplied by 2, 3, 4, and 5.Prediction: The next number will be multiplied by 6. So, itwill be (6)(240) or 1440.Answer: 1440

Your Turn:Make a prediction about the next number based on thepattern.1 1 1 11, , ,4 9 16, 25Answer: The next number will be11or2636

Joke Time What’s a fresh vegetable? One that insults a farmer. What did one knife say to the other? Look sharp! What did one strawberry say to the other? “Look at the jam you’ve gotten us into!”

Ordered Pair xy(x, y) The points form a line. y x –4; x –4, –2, 0, 2, 4 1 2 Your Turn: Generate ordered pairs for the function using the given values for x. Graph the ordered pairs and describe the pattern.