MATH 3162 Review Chapter 13 and Chapter 8
Chapter 13: Exploring ideas of algebra and coordinate geometry
13.3
Exploring graphs of linear equations
13.3.1. Developing techniques for graphing linear equations
13.3.1.1. x-axis
13.3.1.1.1. abscissa
13.3.1.1.2. horizontal axis
13.3.1.2. y-axis
13.3.1.2.1. ordinate
13.3.1.2.2. vertical axis
13.3.1.3. x and y axis intersection
13.3.1.3.1. (0,0)
13.3.1.3.2. origin
13.3.1.3.3. quadrants
13.3.1.3.3.1.
four
13.3.1.3.3.2.
Roman numerals
13.3.1.3.3.3.
top right = I (one) and (+,+)
13.3.1.3.3.4.
top left = II (two) and (-,+)
13.3.1.3.3.5.
bottom left = III (three) and (-,-)
13.3.1.3.3.6.
bottom right = IV (four) and (+,-)
13.3.1.4. coordinate
13.3.1.4.1. ordered pair of numbers
13.3.1.4.2. (x, y)
13.3.1.4.3. (2, 3) is not the same as (3, 2)
13.3.1.5. making a table
13.3.1.5.1. example: y = 2x + 1
x
1
0
-1
2x + 1
2(1) + 1
2(0) + 1
2(-1) + 1
y
3
1
-1
(x, y)
(1,3)
(0,1)
(-1,-1)
y
5
x
-5
0
5
-5
13.3.2. Understanding how graphs and equations are related
13.3.2.1. slope
rise
13.3.2.1.1.
run
y 2  y1
m
13.3.2.1.2.
x 2  x1
13.3.2.1.3. positive slope
13.3.2.1.3.1.
goes up from left to right
13.3.2.1.4. negative slope
13.3.2.1.4.1.
goes down from left to right
13.3.2.2. y-intercept
13.3.2.2.1. where the graph crosses the y-axis
13.3.2.2.2. (0,y)
13.3.2.2.3. In y = mx + b, the y-intercept is (0,b)
13.3.2.3. parallel lines have the same slope: m1 = m2
13.3.2.4. perpendicular lines
13.3.2.4.1. m1 x m2 = -1
13.3.2.5. Slope and the equation of the line
13.3.2.5.1. (y2 – y1) = m(x2 – x1)
y 2  y1
m
13.3.2.5.2.
x 2  x1
13.3.3. Understanding how slope is related to horizontal and vertical lines
13.3.3.1. horizontal lines
13.3.3.1.1. all points on the line in the form (x, n), where n is a constant
13.3.3.1.2. m = 0
13.3.3.2. vertical lines
13.3.3.2.1. all points on the line in the form (n, y), where n is a constant
13.3.3.2.2. m = undefined
Distance Formula:
13.4
d
x 2  x1 2  y 2  y1 2
based on the Pythagorean Theorem
Connecting Algebra and Geometry
13.4.1. Finding the midpoint of a line segment
13.4.1.1. Theorem for coordinates of the midpoint: If the coordinates for the
endpoints of PQ are  x1, y1  for P and  x 2 , y2  for Q, then the
 x  x 2 y1  y 2 
,
coordinates of the midpoint, M, of PQ are  x M , y M    1

2 
 2
13.4.2. Finding the distance between two points
13.4.2.1. Distance Formula: If d is the distance between points  x1, y1  and
 x 2 , y2  , then
d
 x 2  x1    y2  y1 
2
2
13.4.3. Finding the equation of a line
13.4.3.1. Slope-intercept method
13.4.3.1.1. Slope is designated as “m”
y y
m 2 1
13.4.3.1.1.1.
x 2  x1
13.4.3.1.2. Slope-Intercept form of a line
13.4.3.1.2.1.
(0,b) is the y-intercept
13.4.3.1.2.2.
y = mx + b
13.4.3.2. Two point method
13.4.3.3. Given  x1, y1  and  x 2 , y2  , then  y2  y1   m  x 2  x1 
13.4.3.3.1. First, solve for m to obtain the slope of the line
13.4.3.3.2. Next choose  x1, y1  (arbitrary could choose  x 2 , y2  if
desired, but can ONLY choose one point)
Then  y  y1   m  x  x1  and solve for y to put the equation
into slope-intercept form
13.4.3.3.4. Leave fractions in improper form – do NOT convert to mixed
numbers
13.4.3.4. Point slope method
13.4.3.4.1. Given m and  x1, y1  , the equation for a line can be found
13.4.3.3.3.
by:  y  y1   m  x  x1 
13.4.3.4.2. Solve for y to put the equation into slope-intercept form
13.4.3.5. Finding the point of intersection of two lines
13.4.3.5.1. First find the equation of each line: y  m1x  b1 and
y  m2 x  b2
13.4.3.5.2. Next, use substitution to solve for x: m1x  b1  m2 x  b2
13.4.3.5.3. After finding x, choose either original equation to solve for y,
say y  m1x  b1
13.4.3.5.4. The x and y you found are the (x,y) of the point of
intersection between the two lines
13.4.3.5.5. What happens when the slopes are equal?
13.4.3.6. Horizontal and vertical lines
13.4.3.6.1. Horizontal line properties
13.4.3.6.1.1.
The slope of every horizontal line is in the form: y = b,
where (0,b) is the y-intercept
13.4.3.6.1.2.
The slope of every horizontal line is zero
13.4.3.6.2. Vertical line properties
13.4.3.6.2.1.
The slope of every vertical line is in the form: x = t,
where (t,0) is the x-intercept
13.4.3.6.2.2.
The slope of every vertical line is undefined; a vertical
line has no slope
13.4.4. Using coordinate geometry to verify geometric conjectures
13.4.4.1. skip
13.4.5. Developing the equation of a circle
13.4.5.1. Circle equation theorem 1: The standard form for an equation of a
circle with a center at the origin and radius r is: x 2  y 2  r 2
13.4.5.2. Circle equation theorem 2: The standard form for an equation of a
circle with a center at (h, k) and radius r is:  x  h    y  k   r 2
13.4.6. Describing transformations using coordinate geometry
13.4.6.1. skip
2
2
Chapter 8: Analyzing Data
8.1 Types of data displays
8.1.1. Common Forms of Data Display
8.1.1.1. data – numerical or categorical
8.1.1.2. statistics – the science of collecting, classifying, and using data to
interpret the significance of numerical and categorical information
8.1.1.3. data displays – graphs and tables
8.1.1.3.1. Title
8.1.1.3.2. Scale
8.1.1.3.3. Labels
8.1.2. Representing Frequency and Distribution of Data
8.1.2.1. Frequency Tables
8.1.2.1.1. Most common form of displaying data
8.1.2.1.2. Categories are chosen (numerical categories must have equal
groups)
8.1.2.1.3. Tally marks used to do initial counts
8.1.2.1.4. Tally marks are summarized with numbers to give the frequency
8.1.2.1.5. Example p. 393 Table 8.1
8.1.2.2. Line Plots
8.1.2.2.1. Used for data sets of 40 points or less
8.1.2.2.2. Numbers in a set are sometimes called data points
8.1.2.2.3. Usually plotted on a horizontal axis or number line
8.1.2.2.4. The range is determined by the data in the set
8.1.2.2.4.1.
Range – Highest value minus Lowest value:
8.1.2.2.4.2.
H–L=R
8.1.2.2.5. Outliers – values that appear to be outside of the expected
values, sometimes thought of as extreme values
8.1.2.2.6. See figure 8.1 p. 394
8.1.2.3. Stem-and-Leaf Plot
8.1.2.3.1. Usually only two digits used
8.1.2.3.2. Stem is the digit in the tens place
8.1.2.3.3. Leaf is the digit in the units place
8.1.2.3.4. Key MUST accompany ALL plots for interpretation
8.1.2.3.5. Unordered plots – usually formed during data gathering process
8.1.2.3.6. Ordered plots – all numbers in each part of the leaf are placed
from lowest to highest
8.1.2.3.7. Combined stem-and-leaf plots use a single scale with leaves
going out to the left and the right
8.1.3. Representing Variation and Comparing Data
8.1.3.1. Pictograph – small icon or figure is used to represent data values
8.1.3.2. Histogram –ranges of data graphed; bars touch
8.1.3.3. Bar Graph – discrete data graphed; bars do NOT touch
8.1.4. Representing Part-to-Whole Relationships in Data
8.1.4.1. Circle graph or pie chart
8.1.4.2. Whole circle always = 100%
8.2 Data displays that show relationships
8.2.1. Displaying two-variable data
8.2.1.1. Discrete data – data that occurs as whole numbers only: people,
keys, cars, etc.
8.2.1.1.1. Measurable at some finite number of points
8.2.1.1.2. Graphed with a dotted line
8.2.1.2. Continuous data – data that can occur as fractional data: weight,
linear measures, liquid measures, temperatures, etc.
8.2.1.2.1. Measurable at each point in time
8.2.1.2.2. Graphed with a solid line
8.2.1.3. Line graphs
8.2.1.3.1. most common method for showing a relationship between two
variables
8.2.1.3.2. used to describe discrete (dotted line) or continuous data (solid
line)
8.2.2. Scatter plots and trend lines
8.2.2.1.1. Scatter plots
8.2.2.1.1.1.
plotted ordered pairs
8.2.2.1.1.2.
correlation
8.2.2.1.1.2.1. positive = positive slope for line of best fit
8.2.2.1.1.2.2. no correlation = slope is zero
8.2.2.1.1.2.3. negative = negative slope for line of best fit
8.2.2.1.1.2.4. trend line is line that best describes the points present
8.2.2.1.1.2.5. NO causation is implied by correlation
8.3 Describing the Average and Spread of Data
8.3.1. Different types of averages
8.3.1.1. arithmetic mean
x  x 2  x3    xn
8.3.1.1.1. x  1
n
sum all terms
8.3.1.1.2. x 
number of terms
8.3.1.2. median
8.3.1.2.1. put the list of numbers in order (low to high or high to low)
8.3.1.2.2. when there are an odd number of terms, it is the term in the
middle of the list
8.3.1.2.3. when there are an even number of terms, it is the average of the
m  m2
two middle terms: 1
 median
2
8.3.1.3. mode
8.3.1.3.1. if there is an equal number of all elements, then there is no
mode
8.3.1.3.2. value that appears most frequently
8.3.1.3.3. can be one, two (bimodal) or more modes for a given set of data
8.3.1.4. midrange
8.3.1.4.1. mean of the largest and smallest numbers
Lowest  Highest
 midrange
8.3.1.4.2.
2
8.3.2. Numerical descriptions of the spread and distribution of data
8.3.2.1. range: Highest – Lowest = range
8.3.2.2. Quartiles
8.3.2.2.1. 1st quartile: median of the scores below the median of the full
set
8.3.2.2.2. 2nd quartile: the median of the full set
8.3.2.2.3. 3rd quartile: the median of the scores above the median of the
full set
8.3.2.3. Interquartile Range (IQR)
8.3.2.3.1. Q3 - Q1 = IQR
8.3.2.3.2. Outliers found outside of the range
8.3.2.3.2.1.
Q1 – 1.5IQR
8.3.2.3.2.2.
1.5IQR + Q3
8.3.2.3.2.3.
(Q1 – 1.5IQR, Q3 + 1.5IQR)
8.3.3. Graphical descriptions of the spread and distribution of data
8.3.3.1. Box and whisker plots
8.3.3.1.1. whisker = range of data
8.3.3.1.2. box = 50% of data: starts at Q1 and ends at Q3
8.3.3.1.3. Median or Q2 marked inside box
8.3.4. Additional ways to describe the spread and distribution of data
8.3.4.1. Variance = s2
8.3.4.2. Standard deviation = s
8.3.4.3. Outliers
8.3.4.3.1. x  2s
8.3.4.3.2. ( x  2s, x  2s)
8.4. Decision Making with Data
8.4.1. Evaluating data collection procedures
8.4.1.1. Was the sample size adequate?
8.4.1.2. Were subjects randomly assigned to groups?
8.4.1.3. Did the test measure what it was supposed to measure, was it
valid?
8.4.1.4. Were the test results reproducible, was it reliable?
8.4.2. Making valid conclusions from data
8.4.2.1. Beware of vague or undocumented statements of comparison
8.4.2.2. Beware of statements that have percents in their claims
8.4.2.3. Beware of conclusions based on correlations that claim causation
8.4.3. Misleading conclusions from graphs
8.4.3.1. unlike data units
8.4.3.2. unequal scales
8.4.3.3. see 437-438 graphs
8.4.4. Evaluating a data-based conclusion
8.4.4.1. Does the conclusion of the graphic seem reasonable?
8.4.4.2. What is the source of the data? Is it representative of the
population it was taken from?
8.4.4.3. Can the facts be verified or the study replicated? Is it a one time
occurrence?
8.4.4.4. Do the conclusions match you own observations?
8.4.4.5. Is a causal relation erroneously assumed?
8.4.4.6. Are comparisons or percents accurately based?
8.4.4.7. Does the argument presented explain why the opposite claim
doesn’t hold?
8.4.4.8. Are the scales and the units included in the graphic clear and not
misleading?
8.4.4.9. If the graph is showing change, are the units of measurement
appropriately handled?
Summary:
Chapter 13 p. 787: 13.3 and 13.4 only
Chapter 8 p. 455: 8.1-8.4
Review:
Chapter 13 p. 789: 9, 15-28
Chapter 8 p. 457: 1-12