The linear function is popular in economics. It is attractive because it is simple and easy to handle mathematically. It has many important applications.
You are watching: Characteristics of linear functions
Linear functions are those whose graph is a straight line.
A linear function has the following form
y = f(x) = a + bx
A linear function has one independent variable and one dependent variable. The independent variable is x and the dependent variable is y.
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a is the constant term or the y intercept. It is the value of the dependent variable when x = 0.
b is the coefficient of the independent variable. It is also known as the slope and gives the rate of change of the dependent variable.
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Graphing a linear function
To graph a linear function:
1. Find 2 points which satisfy the equation
2. Plot them
3. Connect the points with a straight line
Example:
y = 25 + 5x
let x = 1 then y = 25 + 5(1) = 30
let x = 3 then y = 25 + 5(3) = 40
A simple example of a linear equation
A company has fixed costs of $7,000 for plant and equuipment and variable costs of $600 for each unit of output. What is total cost at varying levels of output?
let x = units of output let C = total cost
C = fixed cost plus variable cost = 7,000 + 600 x
| output | total cost |
| 15 units | C = 7,000 + 15(600) = 16,000 |
| 30 units | C = 7,000 + 30(600) = 25,000 |
Combinations of linear equations
Linear equations can be added together, multiplied or divided.
A simple example of addition of linear equations
C(x) is a cost function
C(x) = fixed cost + variable cost
R(x) is a revenue function
R(x) = selling price (number of items sold)
profit equals revenue less cost
P(x) is a profit function
P(x) = R(x) – C(x)
x = the number of items produced and sold
Data:
A company receives $45 for each unit of output sold. It has a variable cost of $25 per item and a fixed cost of $1600. What is its profit if it sells (a) 75 items, (b)150 items, and (c) 200 items?
| R(x) = 45x | C(x) = 1600 + 25x |
| P(x) = 45x -(1600 + 25x) | |
| = 20x – 1600 |
| let x = 75 | P(75) = 20(75) – 1600 = -100 a loss |
| let x = 150 | P(150) = 20(150) – 1600 = 1400 |
| let x = 200 | P(200) = 20(200) – 1600 = 2400 |
