Graph Review
Throughout this course, you often need to read and draw graph. The purpose of these class notes is to review the basic aspects of graphing that will be used throughout the course. How to read and draw a two variable graph is discussed below. The topics that you should master are:
1) Coordinate System
2) Linear Graph
3) Non-linear Graph
1. Coordinate System
We can use the coordinate system to summarize the relationship between 2 variables as follows.
As far as which variable should be in the y-axis or on the x-axis (if there is causation between two variables, i.e. a change in one variable determines the change in another), the convention is that the y-variable or y-axis is the dependent variable, whereas the x-variable or x-axis is the independent variable. In other words, the change in x will result in the change in y, not the other way around. However, the readers should take the convention with caution because sometimes we do not follow this convention in economics. For example, we put quantity – a dependent variable – in the x-axis and price – an independent variable – in the y-axis. On the other hand, we put consumption – a dependent variable – in the y-axis and income – an independent variable – in the x-axis.
2. Linear Graph
When the two variables have a linear relationship (an increase in one unit of x will result in a constant increase in y at any value of x), we can use the straight line to summarize the relationship between the two as follows:
y = ax + b
Where y = variable in the y-axis
x = variable in the x-axis
a = slope of the linear graph (constant value) =
b = vertical intercept of the linear graph (the value of y when x is 0)
Hence, we can represent the above equation using a linear graph as follows.
slope = a =
Now, let us compare different graphs.
2.1 Different intercepts but same slope
2.2 Upward sloping vs. downward sloping
slope a > 0 à upward sloping slope a’ < 0 à downward sloping
positively related negatively related
directly related inversely related
2.3 Steeper vs. flatter slopes
slope a > a’ and |a| > |a’| slope a < a’ and |a| > |a’|
2.4 Horizontal vs. vertical slopes
slope = slope =
x and y are not related x and y are not related
3. Non-linear Graph
When the two variables have a non-linear relationship (an increase in one unit of x will result in a non-constant increase in y depending on the value of x), we can use a curve instead of a straight line to summarize the relationship between the two where the slope at any given point on the graph is equal to the change in y with respect to the change in x, or the derivative of y with respect to x (dy/dx), and is represented by a tangent line to that point as shown in the graph below.
Do the following Examples 1, 2, 3, 4, 5, 6, and 7.
Example 1: Linear Equation 1
There are two sets of x, y points on a straight line in a two-variable graph with y on the vertical axis and x on the horizontal axis. If one set of points was (0, 75) and the other set of points was (75, 25), the linear equation for the line would be:
a) y = 75 - .66x b) y = 75 - .5x
c) y = 75 - 1.5x d) y = 75 - .33x
Answer: Choose (a). Recall y = ax + b, here we have:
Example 2: Linear Equation 2
In a linear equation for price and quantity supplied, the value of the vertical intercept is 16 and the slope of the line is 5. If the price is $296, the quantity supplied is:
a) 144 b) 150
b) 56 d) 24
Answer: Choose (c). Recall y = ax + b and the convention is that price is the y-axis even though it is an independent variable, whereas quantity is the x-axis even though it is a dependent variable. Thus, we have:
p = 5qs + 16 à 296 = 5qs + 16 à 5qs = 280
qs = 56
Example 3: Linear Equation 3
In a linear equation for price and quantity demanded, the vertical intercept is 1000. If you know that when price is $175, the quantity demanded is 55, then the slope of the line is:
a) 5 b) -5
c) 15 d) -15
Answer: Choose (d). Recall y = ax + b and the convention is that price is the y-axis, whereas quantity is the x-axis. Thus, we have:
p = aqd + 1000 à 175 = 55a + 1000
55a = -825
a = -15
Example 4: Linear Equation 4
In a linear equation relating income and consumption, you know that the intercept is $1,000 and the slope of the line is .4. If income is $20,000, then consumption is:
a) $8,000 b) $9,000
c) $10,000 d) $11,000
Answer: Choose (b). Recall y = ax + b and the convention is that consumption (c) – a dependent variable – is the y-axis, whereas income (I) – an independent variable – is the x-axis. Thus, we have:
c = .4I + 1000 à c = .4 x 20000 + 1000 = 8000 + 1000 = 9000
Example 5: Comparing Slopes
5.1 In line (1) on the above graph, the variable x and y are:
a) nonlinearly related b) positively related
c) negatively related d) inversely related
5.2 In line (3) on the above graph, the variable x and y are:
a) directly related b) negatively related
c) positively related d) nonlinearly related
5.3 The slope of line (2) on the above graph is:
a) 0 b) .66
c) .75 d) 1.50
5.4 The linear equation for line (3) on the above graph is:
a) y = 24 – 1.5x b) y = 16 - .5x
c) y = 24 - .66x d) y = 24 - .75x
Answer: Choose (b), (b), (a), and (a) for 5.1-5.4 respectively. Recall y = ax + b and the convention is that a dependent variable is the y-axis (unless it is price), whereas an independent variable is the x-axis (unless it is quantity). From the graph, we have:
In addition, for line (3) we have:
Example 6: Graph and Table
Refer to the table below that shows the quantities of corn consumers will purchase at various prices (hypothetical data).
Price per bushel Bushels demanded per year
$5 2,000
4 4,000
3 7,000
2 11,000
1 16,000
From an economist’s perspective, which line in the above graph best represents the data in the table?
a) A b) B
c) C d) D
Answer: Choose (b). From the table, we have
Price per bushel Bushels demanded per year
$5 2,000 price ↓ à bushels ↑
4 4,000 negatively related
3 7,000 downward sloping
2 11,000
1 16,000 line B
Example 7: Reading A Table
This question is based on the following four data sets, (1), (2), (3), and (4). In each set, the independent variable is in the left column and the dependent variable is in the right column.
(1) / (2) / (3) / (4)A / B / C / D / E / F / G / H
0 / 6 / 2 / 10 / 0 / 12 / 0 / 3
7 / 5 / 3 / 20 / 1 / 10 / 2 / 5
14 / 4 / 4 / 30 / 2 / 8 / 4 / 7
21 / 3 / 5 / 40 / 3 / 6 / 6 / 9
7.1 There is an inverse relationship between the independent and dependent variable in the above data sets:
a) 1 and 3 b) 2 and 3
c) 3 and 4 d) 2 and 4
7.2 The linear equation for data set 1 above is:
a) B = 7A b) A = .14B
c) B = 6 - 7B d) B = 6 - .14A
7.3 The slope of the line for data set 2 above is:
a) .10 b) .20
c) 5 d) 10
7.4 The vertical intercept is 12 in the above data set:
a) 1 b) 2
c) 3 d) 4
7.5 The linear equation for the above data set 4 is:
a) G = 3 + 2H b) H = 3 – 1G
c) G = 3 + 1H d) H = 3 + 1G
Answer: Choose (a), (d), (d), (c), and (d) for 7.1-7.5 respectively. Recall y = ax + b and the convention is that a dependent variable (in the right columns – B, D, F, and H) is the y-axis (unless it is price), whereas an independent variable (in the left columns – A, C, E, and G) is the x-axis (unless it is quantity). From the table, we have:
A / B / C / D / E / F / G / H
0 / 6 / 2 / 10 / 0 / 12 / 0 / 3
7 / 5 / 3 / 20 / 1 / 10 / 2 / 5
14 / 4 / 4 / 30 / 2 / 8 / 4 / 7
21 / 3 / 5 / 40 / 3 / 6 / 6 / 9
x / y / x / y / x / y / x / y
x ↑ à y ↓
inverse / x ↑ à y ↑
direct / x ↑à y ↓
inverse / x ↓ à y ↑
direct
b = 6 / b = n.a. / b=12 / b = 3
B = 6 - .14A / D = b + 10C / F = 12 - 2E / H = 3 + 1G
The idea here is that we can quickly identify the vertical intercept b from the table by looking for the coordinate where x = 0.
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