Graphing Techniques
and
Interpreting Graphs
8 Rules of Graphing
IV/DV
Graphs show relationships
between variables:
1. Linear (directly proportional)
•
Linear
2. Non-Linear (indirectly proportional)
•
•
•
Inverse
Exponential or Quadratic
Oscillating
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
Linear RelationshipsGraphing Data
(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, Δy, is the difference between
the vertical values of A and B.
The horizontal change, or run, Δx,
is the difference between the
horizontal values of A and B.
y 2  y1
y
rise
slope  m 


x
x2  x1
run
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 Δy/Δx is negative, and the
line slopes downward.
• The y-intercept, b, is the point at which the line crosses the yaxis, and it is the y-value when the value of x is zero.
 y   y2  y1 
m 

 x   x2  x1 
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?
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?
2.
Non-Linear Relationships:
Inverse 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.
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?
Inverse Relationship y = k / x
Example:
As the resistance
increases, the
current decreases.
Can you figure out k?
What are the units of k?
Other Non-Linear Relationships:
Exponential 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
Other Non-Linear Relationships:
Quadratic 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.
Other Non-Linear Relationships:
Oscillating Relationships
Oscillating
relationship:
variables
increase and
decrease about
each other.
Examples:
y = sin x
y = cos x
Graphs show relationships
between variables:
1. Linear (directly proportional)
•
Linear
2. Non-Linear (indirectly proportional)
•
•
•
Inverse
Exponential or Quadratic
Oscillating
Section
Learning Check
1.3
Question 1
Which type of relationship is shown
following graph?
A. Linear
C. Exponential or Quadratic
B. Inverse
D. None of the above
Section
Learning Check
1.3
Answer 1
Answer: B
Reason: In an inverse relationship a hyperbola results when one
variable depends on the inverse of the other.
Section
Learning Check
1.3
Question 2
What is line of best fit?
A. The line joining the first and last data points in a graph.
B. The line joining the two center-most data points in a graph.
C. The line drawn close to all data points as possible.
D. The line joining the maximum data points in a graph.
Section
Learning Check
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.
Section
Section Check
1.3
Question 3
Which relationship can be written as y = mx?
A. Linear relationship
B. Quadratic relationship
C. Parabolic relationship
D. Inverse relationship
Section
Section Check
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.
More Vocabulary:
Interpolation-- finding points between points.
Extrapolation-- finding points beyond the last point.
Most Important: Linear Relationships
Slope
50
m=(40-8)/(50-10)
Mass(g)
40
m=32/40
30
m=0.8 g/cm3
20
10
0
10
20
30
Volume(mL)
40
Interpolation
vs.
Extrapolation
Density
D=m/V
D = Density
m = Mass
V = Volume
Find the density of a sample whose mass is 25.0 g and whose
volume is 82.3 cm3.
Find the mass of a sample whose density is 8.2 g/ cm3 and
whose volume is 52.0 cm3.
Find the volume of a sample whose mass is 250 g and whose
density is 6.3 g/cm3.
IV/DV con’t


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.
IV/DV con’t


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.
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.