Part One:

Introduction to Graphs

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).

Part Two:

Equations and Graphs of Straight Lines

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.

Calculating the Slope

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.

Example

• Points (15, 8) and (10, 7) are on a straight line.

• What is the slope of this line?

Example

• What is the slope of the line given in the graph?

• The slope of this line is 2.

• 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?

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?