Linear Functions

Functions

A function is a mathematical relationship that assigns each value in the domain (input) to exactly one value in the range (output).

• Independent variable: Usually denoted by x.

• Dependent variable: Usually denoted by y, since y depends on x.

• Function test: The same input cannot produce two different outputs.

• Example: (2,5), (3,7), (4,9) is a function; (2,5), (2,8) is not a function.

Linear Functions

A linear function changes at a constant rate and is represented by the equation:

• m: Slope or rate of change

• b: y-intercept or starting value

• Example: ; slope is 5, intercept is 20.

Slope

The slope measures how much the dependent variable changes for each unit change in the independent variable.

• Positive slope: x increases, y increases.

• Negative slope: x increases, y decreases.

• Zero slope: y does not change; function is constant.

• Example: Points (2,10) and (6,30):

The Intercept

The y-intercept is the value of y when x = 0.

• in

• Example: ; intercept is 500.

• If , the function is proportional.

Equation of a Line from Two Points

To find the equation of a line given two points:

1. Find slope:

2. Use point-slope form:

3. Substitute one point and simplify to

4. Example: Points (2,10) and (6,30):

Change in the Dependent Variable

The change in y for a change in x is given by:

• Example: If and increases by 6,

Reading a Linear Function from a Graph

• Check if the graph is a straight line.

• Determine slope (positive, negative, zero).

• Find y-intercept (where line crosses y-axis).

• Choose two points and calculate slope.

• Write equation as .

Linear, Constant, and Proportional Functions

• Linear:

• Constant: (slope )

• Proportional: (intercept )

• Examples: is linear; is constant; is proportional.

Applications of Linear Functions

Demand

The Law of Demand states that as price rises, quantity demanded decreases (negative relationship).

• with

• Example: As price increases, quantity demanded decreases.

Supply

The Law of Supply states that as price rises, quantity supplied increases (positive relationship).

• with

• Example: As price increases, quantity supplied increases.

Equilibrium

Equilibrium occurs when quantity supplied equals quantity demanded.

• Set and solve for price and quantity.

• Example: ,

• Set equal:

• Solve:

• Plug into either function:

Revenue

Revenue is the total income from sales.

• Example: Price = R = 15 \times 100 = 1,500 $

Cost

Total cost includes variable and fixed costs.

• MC: Marginal cost per unit

• F: Fixed cost

• Example:

Profit

Profit is the difference between revenue and cost.

• Example: Revenue = \pi = 2,000 - 1,300 = 700 $

Break-Even

Break-even occurs when profit is zero, i.e., revenue equals cost.

• Example: ,

• Set equal:

• Break-even quantity formula:

Relationships Between Lines

Types of Line Relationships

• Parallel: Same slope, different intercepts.

• Coincident: Identical lines; infinitely many intersection points.

• Perpendicular: Lines intersect at a right angle.

Linear Regression

Line of Best Fit

Linear regression estimates the relationship between an independent variable and a dependent variable using a line of best fit.

• Example: Predicted wage = -0.90 + 0.54(education)

• Interpret the slope and identify variables.

Capital Asset Pricing Model (CAPM)

CAPM Formula

The Capital Asset Pricing Model relates expected return and investment risk.

• Know how to substitute values into the formula.

Summary Table: Key Linear Function Formulas

Exam Readiness Checklist

• Explain why a relationship is or is not a function.

• Identify domain, range, independent, and dependent variables.

• Calculate slope from two points and interpret its meaning.

• Identify and interpret the y-intercept.

• Create from two points or a graph.

• Distinguish demand and supply relationships.

• Solve for equilibrium price and quantity.

• Write revenue and cost functions from word problems.

• Calculate profit and break-even quantity.

• Understand linear regression and CAPM if included.