Post on 29-Dec-2015
5.1 THE 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.
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5.2 Relations
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
5.3 Equations 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 isindependent. So, I can use the same values of x for different equations, but what I get out for y will be different.
5.4 GRAPHING LINEAR EQUATIONS
Linear Equation in Standard FormA 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 + 16
-x -x-x + 9y = 16 -1
Is this a linear equation? If so, what are A, B,and C.In order to identify A, B and C, the equationmust be in standard form and A cannot benegative or A, B, nor C can be fractions.
x - 9y = -16A = 1 B = -9 C = -16
Is 2x2 + 3y = 4 a linear equation? If so, identify A, B and C.
It is not linear because one of the variables has an exponent otherthan 1.Is x = -5 a linear equation? If so, identify A, B, and C.Yes it is. A = 1, B = 0 and C = -5.
5.5 FUNCTIONS
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 functionThis means that everywhere I see x, I willreplace 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 | 7y |12| 14 |16 |18 |20
The difference in the xvalues is +1The difference in the yvalues is +2
Therefore y is 2. So, this would suggest that the function would xbe y = 2x. However, when I put in 3 for x and multiply, I get 6and not 12. So I would add six. Is this consistent. If it is, thenwhat this suggests is the function is y = 2x + 6.
5.7 Measures of Variation
The measures of variation often describes the distribution ofdata.
The range of a set of data is the difference between the greatestand least values of the set.
The quartiles are values that divide the data into four equal parts.Statisticians often use Q1, Q2, Q3.
Q1 (lower quartile) divides the lower half into two equal parts. Q2 is the medianQ3 (upper quartile) divides the upper half of the data into two equal parts.
The difference between the upper quartile Q3 and the lower quartile Q1 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)
ExampleMrs. 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 themedian (Q2), the upper (Q3) and lower (Q1) quartile range andany outliers.First, put data in chronological order.14, 15, 19, 19, 20, 20, 20, 20, 22, 22, 23, 23, 24, 30Then find the median (Q2)14, 15, 19, 19, 20, 20, 20,| 20, 22, 22, 23, 23, 24, 30
20 + 20 = 20 The line separates the data into 2 equal parts 2 with seven pieces on each side of the line.
14, 15, 19, 19, 20, 20, 20, | 20, 22, 22, 23, 23, 24, 30 (Q1) (Q2) (Q3)Find the median of the seven pieces of data on the left and right sides of the line. Q1 and Q3 fall directly on two numbers, soI don’t have to do any extra calculations.
So, Q1 is 19, Q2 is 20 and Q3 is 23. Is there an outlier?The IQR which is Q3 - Q1 = 23 - 19 = 4
1.5(4) = 6.Q1 - 6 = 19 - 6 = 13. Are there any values less than 13? No.Q3 + 6 = 23 + 6 = 29. Are there any values more than 29? Yes.30 is an outlier.