Part One: Introduction to Graphs Mathematics and Economics In economics many relationships are...
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Transcript of Part One: Introduction to Graphs Mathematics and Economics In economics many relationships are...
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.
An individual buyer's demand curve for corn
• The law of demand:
– Consumers will buy more of a product as its price declines.
Demand Curve for Paperback Books
• Demand reflects an individual's willingness to buy various quantities of a good at various prices.
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.
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
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.
• 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.
• 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.
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
• The equation also has an "a" and "b" in it.
• These are constants that help define the relationship between the two variables.
• 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.
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.
• 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.
• 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.
• 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
• 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.
• We can set up an equation to show how the total price of pizza relates to the number of toppings ord
• 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
y(Final Price)
a(Plain)
b(Price of Each
Topping)
x(Number of Toppings)
$ 7.00 $ 7.00 $.75 0$ 7.75 $ 7.00 $.75 1$ 8.50 $ 7.00 $.75 2$ 9.25 $ 7.00 $.75 3$10.00 $ 7.00 $.75 4
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.
• 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).
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.
• Identifying the x-coordinate– Draw a straight line fr
om the point directly to the x-axis.
– The number where the line hits the x-axis is the value of the x-coord
• 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.
Notation for Identifying Points
• Coordinates of point B are (100, 400)
• Coordinates of point D are (400, 100)
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.
• 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.
• Step Three
– The point where these two lines intersect is at the point we are plotting, (200, 300).
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.
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
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.
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.
• 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.
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.
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.
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.
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)
• 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.
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.
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.
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.
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
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.
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.
• 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.
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.
• The greater the slope, the steeper the line.
• Keep in mind, you can only make this comparison between lines on a same graph.
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.
• 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.
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,
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
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.
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.
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
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?