Graphing Techniquesand

Interpreting Graphs

8 Rules of Graphing8 Rules of Graphing

IV/DVIV/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

Graphing Data

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.

run

rise

xx

yy

x

ymslope

12

12

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 y-axis, and it is the y-value when the value of x is zero.

12

12

xx

yy

x

ym

Linear Relationships(Directly Proportional)

Example: Mass vs. Volume: As the volume increases, so does the mass.

Linear Relationships(Directly Proportional)

What is the equation of one of these lines?What are the units for its slope?What is the meaning of the slope?

Example: Mass vs. Length: As the mass increases, the length of the spring increases.

Linear Relationships(Directly Proportional)

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 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 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

y = x2You 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.

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

Which type of relationship is shown following graph?

Question 1

Section

1.3 Learning Check

A. Linear

B. Inverse

C. Exponential or Quadratic

D. None of the above

Learning Check

Answer: B

Answer 1

Section

1.3

Reason: In an inverse relationship a hyperbola results when one variable depends on the inverse of the other.

Learning Check

What is line of best fit?

Question 2

Section

1.3

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.

Learning Check

Answer: C

Answer 2

Section

1.3

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 Check

Which relationship can be written as y = mx?

Question 3

Section

1.3

A. Linear relationship

B. Quadratic relationship

C. Parabolic relationship

D. Inverse relationship

Section Check

Answer: A

Answer 3

Section

1.3

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:More Vocabulary:

Interpolation-- finding points between points.

Extrapolation-- finding points beyond the last point.

Most Important: Linear Relationships

0

10

20

30

40

50

10 20 30 40

Volume(mL)

Mass

(g)

Slope

m=(40-8)/(50-10)

m=32/40

m=0.8 g/cm3

Interpolation

vs.

Extrapolation

DensityDensity

D = m / VD = Densitym = MassV = Volume

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.

IV/DV con’tIV/DV con’t The relationship between the The relationship between the

independent and dependent variables independent and dependent variables is called a function.is called a function.

Ex 1: The longer you walk, the greater Ex 1: The longer you walk, the greater the distance from where you started.the distance from where you started. In this example, the independent variable In this example, the independent variable

is the time walking, and the dependent is the time walking, and the dependent variable is the distance from the starting variable is the distance from the starting point. We can say that the distance point. We can say that the distance covered is a function of time.covered is a function of time.

IV/DV con’tIV/DV con’t

Ex 2: Money earned and hours worked. Ex 2: Money earned and hours worked. In this example, the amount of money In this example, the amount of money

you earn depends on the number of you earn depends on the number of hours you worked. So the independent hours you worked. So the independent variable is the hours worked and the variable is the hours worked and the dependent variable is the money dependent variable is the money earned. Money earned is a function of earned. Money earned is a function of the hours worked.the hours worked.

IV/DV RelationshipsIV/DV RelationshipsIndependent and dependent variables Independent and dependent variables

exist in relationships with one another. exist in relationships with one another. Direct relationship: Both variables Direct relationship: Both variables

increase; on a graph, this line would increase; on a graph, this line would have a positive slope.have a positive slope.

Indirect relationship: One variable Indirect relationship: One variable increases, the other decreases; on a increases, the other decreases; on a graph, this line would have a positive graph, this line would have a positive slope. This is also called an inverse slope. This is also called an inverse relationship.relationship.