Scatter Plots and Linear vs. Nonlinear Relationship
Scatter Plots
Section Overview
● Understand how a line
of best fit can be used to estimate the relationship between variables.
● Draw or identify the
most appropriate line of best fit for a scatter plot.
● Identify when data is
not correlated enough to have a line of best fit.
● Identify from a graph
whether the values of m and c in the equation of the line of best fit y = mx + c
are positive or negative.
● Estimate the value of
data at a point, using a drawn line of best fit.
Key Vocabulary:
Bivariate Data - A data that has two
variables.
Categorical Data - A data that can be grouped by characteristics.
Cluster - Where most of the data is gathered.
Correlation - Relationship between two variables.
Linear Association - Having a constant rate of change (slope).
Line of Best Fit - A straight line that best represents the data on a
scatter plot.
Negative Association - Inverse relationship between two.
Nonlinear Association - Rate of change is not constant; not a straight line.
Outlier - Data that lies “outside” of most of the data; data that is
distinctly separate from other data.
Positive Association - Both variables increase.
Scatter Plot - A graph of plotted points that show the relationship between
two sets of data.
Trend Line - A line on a graph showing the general direction that a group
of points seem to be heading.
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When You Should use a Scatter Plot
Scatter plots’ primary uses are to observe and show relationships
between two numeric variables. The dots in a scatter plot not only report the
values of individual data points, but also patterns when the data are taken as
a whole.
Identification of correlational relationships are common with scatter plots. In
these cases, we want to know, if we were given a particular horizontal value,
what a good prediction would be for the vertical value. You will often see the
variable on the horizontal axis denoted an independent variable, and the
variable on the vertical axis the dependent variable. Relationships between
variables can be described in many ways: positive or negative, strong or weak,
linear or nonlinear.
Types of Correlations:
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The trend line that shows the relationship between two sets of data most
accurately is called the line of best fit.
A graphing calculator or computer program can compute the equation of
the line of best fit using a method called linear regression.
The graphing calculator gives you the correlation coefficient, r,
which tells you how closely the equation models the data.
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When the data points cluster around a line, there is a strong correlation
between the line and the data. So, the nearer r is to 1 or -1, the more
closely the data cluster around the line of best fit.
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A) Calculate the Xmean, Ymean point (mark with an X on the graph)
Rounded to the nearest tenth.
B) Draw the line of best fit, showing the general trend of the line, making
sure the line passes through Xmean, Ymean point
C) Choose one point on the line of best fit, the points may not necessarily be
a data point, and the Xmean, Ymean
E) Use the slope and one of the points to substitute into y = mx
+ b
F) Solve for b.
G) Write the equation of the line in slope-intercept form by substituting m
and b into y = mx + b
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Let’s Practice.
Eyeballing
the line of best fit (practice) | Khan Academy
Linear vs. Nonlinear Relationships
Section Overview
● Use patterns to predict missing values in a table or other
representation of a linear relationship.
● Use and meaningfully represent the concept of slope as a
relationship between the change in one variable compared to the change in
another variable.
● Learn what defines a linear function.
● Represent and understand proportional relationships and direct
proportion.
Key Vocabulary:
Function - A special relationship between values; each of its input
values gives back exactly one output value.
Linear Function - A function that has a constant rate of change.
Nonlinear Function - A function that does not have a constant rate of
change.
Linear Functions
Linear functions are functions that appear as straight lines when
they are graphed.
Because of this, linear functions allow the standard equation of a line, which
is called “slope-intercept form”. Even if an equation doesn't look like it's in
slope-intercept form when you first see it, you can almost always transform it
to get it there, using the Addition and Subtraction Property of Equality and
the Multiplication and Division Property of Equality.
f(x) = mx + b
m = slope
b = y-intercept
Nonlinear Functions
Nonlinear functions are those that are NOT straight lines when they are
graphed.
Because of this, nonlinear functions have many forms. They do not
follow the equation of a line.
There are many standard nonlinear functions, such as quadratic equations.
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Identifying Functions from
Tables
Example: Does each table represent a linear or nonlinear function?
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As x increases by 3, y decreases by 8. The rate of change is
constant. So, the function is linear.
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As x increases by 2,
y increases by different amounts. The rate of change is not constant.
Example 1: Account A earns simple
interest. Account B earns compound interest. The table shows the balance for 5
years. Graph the data and compare graphs.
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Both graphs show that the balances are positive and increasing.
The balance of Account A has a constant rate of
change or $10. So, the function representing Account A is linear.
The balance of Account B increases by different amounts each year. Because the
rate of change is not constant, the function representing the balance of
Account B is nonlinear.
Let’s Practice
1) Which of the following equations is NOT linear?
| A) y = 3x + 5 B) x + y = 20 C) y = x² D) (1/2)x + (1/4)y = 3/4 |
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Which point also lies on the graph of this function?
| A) (3, 3) B) (-2, -4) C) (2, 1) D) (-4, -6) |
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3) Which of the following four functions is (are) linear functions?
Function A: y = x² - 2 |
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4) Does the equation y - 2 = 1/2(x - 3) define y as a linear function of x?
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5) Does the table represent a linear function?
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