CHAPTER 5 THE COORDINATE PLANE THE BEGINNING!!

5.1THE COORDINATE PLANE Points are located in reference to two perpendicular number lines called axes. The axes intersect at their zero points, a point called the origin. The horizontal number line, called the x-axis, and the vertical number line, called the y-axis, divide the plane into four quadrants numbered counter-clockwise.

The plane containing the x and y axes is called the coordinate plane or the Cartesian plane. Points in the coordinate plane are named by ordered pairs of the form (x, y). The first number, or x-coordinate, corresponds to the numbers on the x axis. The second number, or y-coordinate, corresponds to the numbers on the y-axis. The ordered pair for the origin is (0,0). To graph an ordered pair means to draw a dot at the point on the coordinate plane that corresponds to the ordered pair.

5.2Relations A relation is a set of ordered pairs. Relations can be represented by 1) ordered pairs, 2) tables, 3) graphs and 4) mapping. The domain of a relation is the set of all first coordinates from the ordered pairs in the relation. The range of the relation is the set of all second coordinates from the ordered pairs. For example: A = {(2,3), (-2,4), (-1,0), (-4,-5)} A relation {2, -2, -1, -4} The domain {3, 4, 0, -5} The range

The inverse of any relation is obtained by switching the coordinates in each ordered pair. For Example: A = {(2,3), (-2,4), (-1,0), (-4,-5)}A relation B= {(3,2), (4,-2), (0,-1), (-5,-4)} The inverse relation

5.3Equations as Relations If a true statement results when the numbers in an ordered pair are substituted into an equation in two variables, then the ordered pair is a solution of the equation. The domain contains values represented by the independent variable. The range contains the corresponding values represented by the dependent variable.

When you solve an equation for a given variable, that variable becomes the dependent variable because it depends upon the domain values chosen for the other variable. First, solve the equation for y. Put in values for x to get out values for y. Plot the points and draw the line. It does not matter what values of x to put in because it is independent. So, I can use the same values of x for different equations, but what I get out for y will be different.

5.4GRAPHING LINEAR EQUATIONS Linear Equation in Standard Form A linear equation is an equation that can be written in the form Ax + By = C, where A, B, and C are any real numbers, and A and B are not both zero. Linear equations may contain one or two variables with no variable having an exponent other than 1.

For example: 9y = x x -x + 9y = 16 Is this a linear equation? If so, what are A, B, and C. In order to identify A, B and C, the equation must be in standard form and A cannot be negative or A, B, nor C can be fractions. x - 9y = -16 A = 1B = -9 C = -16 Is 2x 2 + 3y = 4 a linear equation? If so, identify A, B and C. It is not linear because one of the variables has an exponent other than 1. Is x = -5 a linear equation? If so, identify A, B, and C. Yes it is. A = 1, B = 0 and C = -5.

5.5FUNCTIONS A function is a relation in which each element of the domain is paired with exactly one element of the range. There are several methods in determining whether a relation is a function: a table, a graph, mapping, and the vertical line test. The vertical line test is performed with a pencil and if a vertical line (the pencil in this case) does not pass through more than one point of the graph of a relation, then the relation is a function.

Equations written in the form f(x) = 3x – 7 is in the form of functional notation. Example: f(x) = 4x + 2 Evaluate f(-4) f(x) is the name of the function This means that everywhere I see x, I will replace it with a -4. 4(-4) + 2 = -14 Therefore, f(-4) = -14

5.6 Writing Equations from Patterns Sometimes, equations can be written by having a graph of a line and looking at some ordered pairs on the line. For example: If we had the following ordered pairs x | 3 | 4 | 5 | 6 | 7 y |12| 14 |16 |18 |20 The difference in the x values is +1 The difference in the y values is +2 Therefore y is 2. So, this would suggest that the function would x be y = 2x. However, when I put in 3 for x and multiply, I get 6 and not 12. So I would add six. Is this consistent. If it is, then what this suggests is the function is y = 2x + 6.

5.7 Measures of Variation The measures of variation often describes the distribution of data. The range of a set of data is the difference between the greatest and least values of the set. The quartiles are values that divide the data into four equal parts. Statisticians often use Q 1, Q 2, Q 3. Q 1 (lower quartile) divides the lower half into two equal parts. Q 2 is the median Q 3 (upper quartile) divides the upper half of the data into two equal parts.

The difference between the upper quartile Q 3 and the lower quartile Q 1 of a set of data is called the interquartile range (IQR). It represents the middle half, or 50% of the data in the set. An outlier is defined as any element of a set of data that is at least 1.5 interquartile ranges greater than the upper quartile or less than the lower quartile. Outliers can be calculated as follows: 1.5(IQR) + Q3 Q1 – 1.5(IQR)

Example Mrs. Kollar gave a quiz in her statistics class. The scores were 23, 30, 22, 20, 20, 14, 15, 19, 19, 20, 23, 20, 22, and 24. Find the median (Q 2 ), the upper (Q 3 ) and lower (Q 1 ) quartile range and any outliers. First, put data in chronological order. 14, 15, 19, 19, 20, 20, 20, 20, 22, 22, 23, 23, 24, 30 Then find the median (Q 2 ) 14, 15, 19, 19, 20, 20, 20,| 20, 22, 22, 23, 23, 24, = 20The line separates the data into 2 equal parts 2with seven pieces on each side of the line.

14, 15, 19, 19, 20, 20, 20, | 20, 22, 22, 23, 23, 24, 30 (Q 1 ) (Q 2 ) (Q 3 ) Find the median of the seven pieces of data on the left and right sides of the line. Q 1 and Q 3 fall directly on two numbers, so I don’t have to do any extra calculations. So, Q 1 is 19, Q 2 is 20 and Q 3 is 23. Is there an outlier? The IQR which is Q 3 - Q 1 = = 4 1.5(4) = 6. Q = = 13. Are there any values less than 13? No. Q = = 29. Are there any values more than 29? Yes. 30 is an outlier.