2 Goals Goal To use tables, equations, and graphs to describe relationships.
3 Vocabulary Solution of an equation Inductive reasoning
4 Review: Graphing in the Coordinate Plane The coordinate plane is formed by the intersection of two perpendicular number lines called axes. The point of intersection, called the origin, is at 0 on each number line. The horizontal number line is called the x-axis, and the vertical number line is called the y-axis.
5 Graphing in the Coordinate Plane Points on the coordinate plane are described using ordered pairs. An ordered pair consists of an x-coordinate and a y-coordinate and is written (x, y). Points are often named by a capital letter. Reading Math The x-coordinate tells how many units to move left or right from the origin. The y-coordinate tells how many units to move up or down.
6 Example: Graphing in the Coordinate Plane Graph each point. A. T( 4, 4) Start at the origin. Move 4 units left and 4 units up. 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 units down. T( 4, 4) V( 2, 3) U(0, 5)
7 Your Turn: Graph each point. A. R(2, 3) Start at the origin. Move 2 units right and 3 units down. B. S(0, 2) Start at the origin. Move 2 units up. C. T( 2, 6) Start at the origin. Move 2 units left and 6 units up. T( 2,6) S(0,2) R(2, 3)
8 Graphing in the Coordinate Plane The axes divide the coordinate plane into four quadrants. Points that lie on an axis are not in any quadrant.
9 Example: Locating Points Name the quadrant in which each point lies. A. E Quadrant ll B. F no quadrant (y-axis) C. G Quadrant l E H y F G x D. H Quadrant lll
10 Your Turn: Name the quadrant in which each point lies. A. T no quadrant (y-axis) B. U Quadrant l C. V Quadrant lll D. W Quadrant ll V W y T U x
11 The Rectangular Coordinate System SUMMARY: The Rectangular Coordinate System Composed of two real number lines one horizontal (the x-axis) and one 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 are called the coordinates of the point. We call the x the x-coordinate and the y the y-coordinate. If both x and y are positive, the point lies in quadrant I; if x is negative, but y is positive, the point lies in quadrant II; if x is negative and y is negative, the point lies in quadrant III; if x is positive 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-axis have an x-coordinate of 0.
12 Equation in Two Variables An equation in two variables, x and y, is a statement in which the algebraic expressions involving x and y are equal. The expressions are called sides of the equation. x + y = 15 x 2 2y 2 = 4 y = 1 + 4x Any values of the variables that make the equation a true statement are said to be solutions of the equation. x + y = 15 The ordered pair (5, 10) is a solution of the equation = = 15
13 Solutions to Equations Example: Determine if the following ordered pairs satisfy the equation 2x + y = 5. a.) (2, 1) b.) (3, 4) 2x + y = 5 2x + y = 5 2(2) + (1) = 5 2(3) + ( 4) = = ( 4) = 5 5 = 5 True 2 = 5 False (2, 1) is a solution. (3, 4) is not a solution.
14 Equation in Two Variables An equation that contains two variables can be used as a rule to generate ordered pairs. When you substitute a value for x, you generate a value for y. The value substituted for x is called the input, and the value generated for y is called the output. Output Input y = 10x + 5
15 Table of Values Use the equation y = 6x + 5 to complete the table and list the ordered pairs that are solutions to the equation. x = 2 y = 6x + 5 y = 6( 2) + 5 y = y = 77 x y (x, y) ( 2, 7) (0, 5) (2, 17) x = 0 y = 6x + 5 y = 6(0) + 5 y = y = 5 x = 2 y = 6x + 5 y = 6(2) + 5 y = y = 17
16 Example: Application An engraver charges a setup fee of $10 plus $2 for every word engraved. Write a rule for the engraver s fee. Write ordered 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 the number of words engraved. Engraver s fee is $10 plus $2 for each word y = x y = x
17 Writing Math The engraver s fee is determined by the number of words in the engraving. So the number of words is the input and the engraver s fee is the output.
18 Example: Solution Number of Words Engraved Rule Charges Ordered Pair x (input) y = x y (output) (x, y) 5 y = (5) 20 (5, 20) 10 y = (10) 30 (10, 30) 15 y = (15) 40 (15, 40) 20 y = (20) 50 (20, 50)
19 Your Turn: What if? The caricature artist increased his fees. He now charges a $10 set up fee plus $20 for each person in the picture. Write a rule for the artist s new fee. Find the artist 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 of people in the picture. Artist s fee is $10 plus $20 for each person y = x y = x
20 Solution: Number of People in Picture Rule Charges Ordered Pair x (input) y = x y (output) (x, y) 1 y = (1) 30 (1, 30) 2 y = (2) 50 (2, 50) 3 y = (3) 70 (3, 70) 4 y = (4) 90 (4, 90)
21 Graphing Ordered Pairs When you graph ordered pairs generated by a function, they may create a pattern.
22 Example: Graphing Ordered Pairs Generate ordered pairs for the function using the given values for x. Graph the ordered pairs and describe the pattern. y = 2x + 1; x = 2, 1, 0, 1, 2 Input Output Ordered Pair x y (x, y) 2 2( 2) + 1 = 3 ( 2, 3) ( 1) + 1 = 1 2(0) + 1 = 1 2(1) + 1 = 3 2(2) + 1 = 5 ( 1, 1) (0, 1) (1, 3) (2, 5) The points form a line.
23 Your Turn: Generate ordered pairs for the function using the given values for x. Graph the ordered pairs and describe the pattern. y = 1 x 4; x = 4, 2, 0, 2, 4 2 Input Output Ordered Pair x y (x, y) = 6 ( 4, 6) = = = = 2 ( 2, 5) (0, 4) (2, 3) (4, 2) The points form a line.
24 Definition Inductive Reasoning is the process of reaching a conclusion based on an observed pattern. Can be used to predict values based on a pattern.
25 Inductive Reasoning Moves from specific observations to broader generalizations or predictions from a pattern. Steps: 1. Observing data. 2. Detect and recognizing patterns. 3. Make generalizations or predictions from those patterns. Observation Pattern Predict
26 Example: Inductive Reasoning Make a prediction about the next number based on the pattern. 2, 4, 12, 48, 240 Find a pattern: The numbers are multiplied by 2, 3, 4, and 5. Prediction: The next number will be multiplied by 6. So, it will be (6)(240) or Answer: 1440
27 Your Turn: Make a prediction about the next number based on the pattern ,,, ,25 Answer: The next number will be 1 1 or
28 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!
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