Graphing Techniques and Interpreting Graphs

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Graphing TechniquesandInterpreting Graphs
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8 Rules of GraphingIV/DV
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Graphs show relationships between variables

• Linear (directly proportional)
• Linear
• Non-Linear (indirectly proportional)
• Inverse
• Exponential or Quadratic
• Oscillating

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1. Linear Relationships(Directly Proportional)

• When the line of best fit is linear (a straight
  line), the variables are directly proportional to
  each other.
• The equation y mx b defines the line.
• m represents slope
• b represents the y-intercept
• As one variable increases, so does the other.

y mx b
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Graphing Data
Linear Relationships(Directly Proportional)

• The slope is the ratio of the vertical change to
  the horizontal change. To find the slope, select
  two points, A and B, far apart on the line. The
  vertical change, or rise, ?y, is the difference
  between the vertical values of A and B. The
  horizontal change, or run, ?x, is the difference
  between the horizontal values of A and B.

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Linear Relationships(Directly Proportional)

• Finding the Slope on a Linear Graph
• Pick two points that are far apart on the line.
  They need not always be data points.
• If y gets smaller as x gets larger, then ?y/?x is
  negative, and the line slopes downward.
• The y-intercept, b, is the point at which the
  line crosses the y-axis, and it is the y-value
  when the value of x is zero.

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Linear Relationships(Directly Proportional)
Example Mass vs. Volume As the volume
increases, so does the mass.
What is the equation of one of these lines? What
are the units for its slope? What is the meaning
of the slope?
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Linear Relationships(Directly Proportional)
Example Mass vs. Length As the mass increases,
the length of the spring increases.
Equation of the line? Slope of the line? Units of
the slope?
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2. Non-Linear RelationshipsInverse Relationship

• y k/x
• As one variable increases, the other variable
  decreases
• k is called a constant
• k is whatever number fixes the equation and
  makes it true for x and y.

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Inverse Relationship y k / x
Example As the speed increases, the time for the
trip decreases.
Can you figure out k? What are the units of k?
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Inverse Relationship y k / x
Example As the resistance increases, the
current decreases.
Can you figure out k? What are the units of k?
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Other Non-Linear RelationshipsExponential
Relationship

• Examples
• y x2
• y x3
• y x -5
• y x 1/2

You cannot tell for sure whether a function is
quadratic or exponential just from the graph. 
There are other functions whose graphs look like
quadratics and exponentials.
y x2
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Other Non-Linear RelationshipsQuadratic
Relationship

• A quadratic relationship can be represented by
  the following equation

Shape is a parabola has a maximum or a minimum,
and is symmetric about a vertical axis.  Often
looks U Shaped, but can be deceptive for
example, if small portions are magnified they can
look like straight lines.
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Other Non-Linear RelationshipsOscillating
Relationships

• Oscillating relationship variables increase
  and decrease about each other.
• Examples
• y sin x
• y cos x

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Graphs show relationships between variables

• Linear (directly proportional)
• Linear
• Non-Linear (indirectly proportional)
• Inverse
• Exponential or Quadratic
• Oscillating

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Learning Check
Section
1.3
Question 1

• Which type of relationship is shown following
  graph?

• Linear
• Inverse

• Exponential or Quadratic
• None of the above

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Learning Check
Section
1.3
Answer 1

• Answer B

Reason In an inverse relationship a hyperbola
results when one variable depends on the inverse
of the other.
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Learning Check
Section
1.3
Question 2

• What is line of best fit?

• The line joining the first and last data points
  in a graph.
• The line joining the two center-most data points
  in a graph.
• The line drawn close to all data points as
  possible.
• The line joining the maximum data points in a
  graph.

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Learning Check
Section
1.3
Answer 2

• Answer C

Reason The line drawn closer to all data points
as possible, is called a line of best fit. The
line of best fit is a better model for
predictions than any one or two points that help
to determine the line.
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Section Check
Section
1.3
Question 3

• Which relationship can be written as y mx?

• Linear relationship
• Quadratic relationship
• Parabolic relationship
• Inverse relationship

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Section Check
Section
1.3
Answer 3

• Answer A

Reason Linear relationship is written as y mx
b, where b is the y intercept. If y-intercept
is zero, the above equation can be rewritten as y
mx.
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More Vocabulary
Interpolation-- finding points between
points. Extrapolation-- finding points beyond
the last point.
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Most Important Linear Relationships
Slope m(40-8)/(50-10) m32/40 m0.8 g/cm3
Interpolation vs. Extrapolation
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Density
D Densitym MassV Volume
D m / V
Find the density of a sample whose mass is 25.0 g
and whosevolume is 82.3 cm3. Find the mass of a
sample whose density is 8.2 g/ cm3 andwhose
volume is 52.0 cm3. Find the volume of a sample
whose mass is 250 g and whosedensity is 6.3
g/cm3.
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IV/DV cont

• The relationship between the independent and
  dependent variables is called a function.
• Ex 1 The longer you walk, the greater the
  distance from where you started.
• In this example, the independent variable is the
  time walking, and the dependent variable is the
  distance from the starting point. We can say
  that the distance covered is a function of time.

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IV/DV cont

• Ex 2 Money earned and hours worked.
• In this example, the amount of money you earn
  depends on the number of hours you worked. So
  the independent variable is the hours worked and
  the dependent variable is the money earned.
  Money earned is a function of the hours worked.

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IV/DV Relationships

• Independent and dependent variables exist in
  relationships with one another.
• Direct relationship Both variables increase on
  a graph, this line would have a positive slope.
• Indirect relationship One variable increases,
  the other decreases on a graph, this line would
  have a positive slope. This is also called an
  inverse relationship.