SECTION 1'3 GRAPHING DATA

1
SECTION 1.3 GRAPHING DATA

•  Objectives
• Graph the relationship between Independent and
  Dependent Variables.
• Interpret Graphs.
• Recognize common relationships in graphs.

2
INTRO

• A well designed graph can convey information
  quickly and simply.
• In this section you will develop graphing
  techniques that will enable you to display,
  analyze, and model data.

3
IDENTIFYING VARIABLES

• When you perform an experiment, it is important
  to change only one variable at a time.
•  
• Variable - any factor that might affect the
  behavior of an experimental setup.
•  
• Independent Variable - is the factor that is
  changed or manipulated during the experiment.
•  
• Dependent Variable - is the factor that depends
  on the independent variable.
• One way to analyze data is to use a line graph
  (see Fig. 1-15). This shows how the dependent
  variable changes with the independent variable.

4
IDENTIFYING VARIABLES

• Line of Best Fit a line that best passes
  through or near graphed data. It is used to
  describe data and predict where new data will
  appear on the graph.
•  
• Problem Solving Strategy p. 16
• The Independent Variable is plotted on the
  Horizontal (x) axis and The Dependent Variable is
  plotted on the Vertical (y) axis.

5
LINEAR RELATIONSHIPS

• The line of best fit may be called the Curve of
  Best Fit for non-linear graphs.
•  
• Linear Relationship - When the line of best fit
  is a straight line, as in the figure 1-16, the
  dependent variable varies linearly with the
  independent variable.
•  
• The Linear Relationship can be written as an
  equation
• as y mx b
• Slope - 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

• Slope m y2 y1 Rise ?y
• x2 x1 Run
  ?x
•  
•  
• If y gets smaller as x gets larger, then ?y/?x is
  negative, and the line slopes downward.
•  
• If y gets larger as x gets larger then slopes
  upward (positive).
• 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.

7
NONLINEAR RELATIONSHIPS

• When the graph is not a straight line, it means
  that the relationship between the dependent
  variable and the independent variable is not
  linear.
•  
• There are many types of nonlinear relationships
  in science. Two of the most common are the
  quadratic and inverse relationships.
•  
• Quadratic Relationship - exists when one variable
  depends on the square of another. The resulting
  graph is a Parabola. It can be represented by
  the following equation
• y ax2 bx c
• Inverse Relationship exists when a variable
  depends on the inverse of another variable. The
  resulting graph is a Hyperbola. It can be
  represented by the following equation
• y a / x

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NONLINEAR RELATIONSHIPS

• There are various mathematical models available
  apart from the three relationships you have
  learned. Examples include sinusoidsused to
  model cyclical phenomena exponential growth and
  decayused to study radioactivity.
• Combinations of different mathematical models
  represent even more complex phenomena.

9
PREDICTING VALUES

• Relations, either learned as formulas or
  developed from graphs, can be used to predict
  values you have not measured directly.
•  
• It is important to decide how far you can
  extrapolate (to estimate to values outside the
  known range) from the data you have.
• Physicists use models to accurately predict how
  systems will behave what circumstances might
  lead to a solar flare, how changes to a circuit
  will change the performance of a device, or how
  electromagnetic fields will affect a medical
  instrument.

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QUESTION 1

• Which type of relationship is shown in the
  following graph?
• Inverse

11
QUESTION 2

• What is a line of best fit?
• 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.

12
QUESTION 3

• Which relationship can be written as y mx?
• 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