Part One:

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• Part One
• Introduction to Graphs

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Mathematics and Economics

• In economics many relationships are represented
  graphically.
• Following examples demonstrate the types of
  skills you will be required to know and use in
  introductory economics courses.

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An individual buyer's demand curve for corn

• The law of demand
• Consumers will buy more of a product as its price
  declines.

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Demand Curve for Paperback Books

• Demand reflects an individual's willingness to
  buy various quantities of a good at various
  prices.

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The concepts you will learn in this section are

• Constant vs. variable.
• Dependent vs. independent variable.
• x and y axes.
• The origin on a graph.
• x and y coordinates of a point.
• Plot points on a graph.

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Variables, Constants, andTheir Relationships

• After reviewing this unit, you will be able to
• Define the terms constant and variable.
• Identify whether an item is a constant or a
  variable.
• Identify whether an item is a dependent or
  independent variable

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Variables and Constants

• Characteristics or elements such as prices,
  outputs, income, etc., are measured by numerical
  values.
• The characteristic or element that remains the
  same is called a constant.
• For example, the number of donuts in a dozen is a
  constant.

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• Some of these values can vary.
• The price of a dozen donuts can change from 2.50
  to 3.00.
• We call these characteristics or elements
  variables.

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• Which of the following are variables and which
  are constants?
• The temperature outside your house.
• The number of square feet in a room that is 12 ft
  by 12 ft.
• The noise level at a concert.

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Relationships Between Variables

• We express a relationship between two variables
  by stating the following The value of the
  variable y depends upon the value of the variable
  x.
• We can write the relationship between variables
  in an equation.
• y a bx

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• The equation also has an "a" and "b" in it.
• These are constants that help define the
  relationship between the two variables.

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• y a bx
• In this equation the y variable is dependent on
  the values of x, a, and b. The y is the dependent
  variable.
• The value of x, on the other hand, is independent
  of the values y, a, and b. The x is the
  independent variable.

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An Example ...

• A pizza shop charges 7 dollars for a plain pizza
  with no toppings and 75 cents for each additional
  topping added.
• The total price of a pizza (y) depends upon the
  number of toppings (x) you order.

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• Price of a pizza is a dependent variable and
  number of toppings is the independent variable.
• Both the price and the number of toppings can
  change, therefore both are variables.

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• The total price of the pizza also depends on the
  price of a plain pizza and the price per topping.
  
• The price of a plain pizza and the price per
  topping do not change, therefore these are
  constants.

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• The relationship between the price of a pizza and
  the number of toppings can be expressed as an
  equation of the form
• y a bx

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• If we know that x (the number of toppings) and y
  (the total price) represent variables, what are a
  and b?
• In our example, "a" is the price of a plain pizza
  with no toppings and "b" is the price of each
  topping.
• They are constant.

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• We can set up an equation to show how the total
  price of pizza relates to the number of toppings
  ord

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• If we create a table of this particular
  relationship between x and y, we'll see all the
  combinations of x and y that fit the equation.
  For example, if plain pizza (a) is 7.00 and
  price of each topping (b) is .75, we get
• y 7.00 .75x

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Graphs

• After reviewing this unit you will be able to
• Identify the x and y axes.
• Identify the origin. on a graph.
• Identify x and y coordinates of a point.
• Plot points on a graph.

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• A graph is a visual representation of a
  relationship between two variables, x and y.
• A graph consists of two axes called the x
  (horizontal) and y (vertical) axes.
• The point where the two axes intersect is called
  the origin. The origin is also identified as the
  point (0, 0).

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Coordinates of Points

• A coordinate is one of a set of numbers used to
  identify the location of a point on a graph.
• Each point is identified by both an x and a y
  coordinate.

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• Identifying the x-coordinate
• Draw a straight line from the point directly to
  the x-axis.
• The number where the line hits the x-axis is the
  value of the x-coord

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• Identifying the y-coordinate
• Draw a straight line from the point directly to
  the y-axis.
• The number where the line hits the axis is the
  value of the y-coordinate.

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Notation for Identifying Points

• Coordinates of point B are (100, 400)
• Coordinates of point D are (400, 100)

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Plotting Points on a Graph

• Step One
• First, draw a line extending out from the x-axis
  at the x-coordinate of the point. In our example,
  this is at 200.

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• Step Two
• Then, draw a line extending out from the y-axis
  at the y-coordinate of the point. In our example,
  this is at 300.

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• Step Three
• The point where these two lines intersect is at
  the point we are plotting, (200, 300).

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• Part Two
• Equations and Graphs of Straight Lines

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Economics and Linear Relationships

• One of the most basic types of relationships is
  the linear relationship.
• Many graphs in economics will display linear
  relationships, and you will need to use graphs to
  make interpretations about what is happening in a
  relationship.

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Inverse relationship between ticket prices and
game attendance

• Two sets of data which are negatively or
  inversely related graph as a downsloping line.
• The slope of this line is -1.25

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Budget lines for 600 income with various prices
for asparagus

• As the price of asparagus rises, less and less
  can be purchased if the entire budget is spent on
  asparagus.

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You will learn in this section to...

• Draw a graph from a given equation.
• Determine whether a given point lies on the graph
  of a given equation.
• Define slope.
• Calculate the slope of a straight line from its
  graph.

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• Be able to identify if a slope is positive,
  negative, zero, or infinite.
• Identify the slope and y-intercept from the
  equation of a line.
• Identify y-intercept from the graph of a line.
• Match a graph with its equation.

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Equations and Their Graphs

• After reviewing this unit, you will be able to
• Draw a graph from a given equation.
• Determine whether a given point lies on the graph
  of a given equation.

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Graphing an Equation

• Generate a list of points for the relationship.
• Draw a set of axes and define the scale.
• Plot the points on the axes.
• Draw the line by connecting the points.

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1. Generate a list of points for the relationship

• In the pizza example, the equation is y 7.00
  .75x.
• You first select values of x you will solve for.
• You then substitute these values into the
  equation and solve for they values.

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2. Draw a set of axes and define the scale

• Once you have your list of points you are ready
  to plot them on a graph.
• The first step in drawing the graph is setting up
  the axes and determining the scale.
• The points you have to plot are
• (0, 7.00), (1, 7.75), (2, 8.50), (3, 9.25), (4,
  10.00)

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• Notice that the x values range from 0 to 4 and
  the y values go from 7 to 10.
• The scale of the two axes must include all the
  points.
• The scale on each axis can be different.

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3. Plot the points on the axes

• After you have drawn the axes, you are ready to
  plot the points.
• Below we plot the points on a set of axes.

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4. Draw the line by connecting the points

• Once you have plotted each of the points, you can
  connect them and draw a straight line.

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Checking a Point in the Equation

• If, by chance, you have a point and you wish to
  determine if it lies on the line, you simply go
  through the same process as generating points.
• Use the x value given in the point and insert it
  into the equation.
• Compare the y value calculated with the one given
  in the point.

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Example

• Does point (6, 10) lie on the line y 7.00
  .75x given in our pizza example?
• To determine this, we need to plug the point (6,
  10) into the equation.
• The point with an x value of 6 that does lie on
  the line is (6, 11.5).
• This means that the point (6, 10) does not lie on
  our line

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Slope

• After reviewing the unit you will be able to
• Define slope.
• Calculate the slope of a straight line from its
  graph.
• Identify if a slope is positive, negative, zero,
  or infinite.
• Identify the slope and y-intercept from the
  equation of a line.
• Identify the y-intercept from the graph of a
  line.

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What is Slope?

• The slope is used to tell us how much one
  variable (y) changes in relation to the change of
  another variable (x).
• This can also be written in the form shown on the
  right.

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• As you may recall, a plain pizza with no toppings
  was priced at 7 dollars.
• As you add one topping, the cost goes up by 75
  cents.

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Calculating the Slope
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Three steps in calculating the slope of a
straight line

• Step One Identify two points on the line.
• Step Two Select one to be (x1, y1) and the other
  to be (x2, y2).
• Step Three Use the slope equation to calculate
  slope.

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Example

• Points (15, 8) and (10, 7) are on a straight
  line.
• What is the slope of this line?

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Example

• What is the slope of the line given in the graph?
• The slope of this line is 2.

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• The greater the slope, the steeper the line.
• Keep in mind, you can only make this comparison
  between lines on a same graph.

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The Sign of Slope

• If the line is sloping upward from left to right,
  so the slope is positive ().
• In our pizza example, as the number of toppings
  we order (x) increases, the total cost of the
  pizza (y) also increases.

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• If the line is sloping downward from left to
  right, so the slope is negative (-).
• For example, as the number of people that quit
  smoking (x) increases, the number of people
  contracting lung cancer (y) decreases.

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Equation of a Line

• The equation of a straight line is given on the
  right. In this equation
• "b" is the slope of the line, and
• "a" is the y-intercept,

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Equation for Pizza Example

• the equation for our pizza example is
• y 7.00 .75 x
• The slope of the line tells us how much the cost
  of a pizza changes as the number of toppings
  change

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y-intercept

• In the equation y a bx, the constant labeled
  "a" is called the y-intercept.
• The y-intercept is the value of y when x is equal
  to zero.

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y-intercept of Pizza Example

• The equation of the relationship is given by y
  7.00 .75 x.
• The y-intercept occurs when there are no
  additional toppings (x 0), which is the price
  of a plain pizza, or 7.00.

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Matching a Graph of a Straight Line with Its
Equation

• After reviewing this unit you will be able to
• Match a graph with its equation

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Matching Using Slope and y-intercept

• We can prove that this is the graph of the
  equation y 2x 10 by checking for two things
  
• Does the line cross the y-axis at 10?
• Is the slope of the line on the graph 2?

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Example

• Consider the following graph at the right.
• Is the equation of the line shown in the graph
  above
• y 4 - 6 x, or
• y 6 - (1/4) x?