Section 2.1 - Functions and Graphs

Linear Functions

• Definition: Linear functions are mathematical functions that result in a straight line when graphed.

• Standard Form: Linear functions are often written as

  • ( y = f(x) = mx + b )

  • Where ( m ) is the slope and ( b ) is the y-intercept.

Graphing Linear Functions

• Process: To graph a linear function, identify two points on the line.

  • Example: For the function ( y = 2x - 3 ):

    • Choose ( x = 0 ): ( y = 2(0) - 3 = -3 \Rightarrow A(0, -3) )

    • Choose ( x = 2 ): ( y = 2(2) - 3 = 1 \Rightarrow B(2, 1) )

    • Plot points A and B and connect them with a straight line.

• Slope-Intercept Form: Useful for quickly identifying the slope and y-intercept of the line.

Slope of Linear Functions

• Definition: The slope ( m ) of a linear function characterizes the steepness of the line, calculated as ( m = \frac{y_2 - y_1}{x_2 - x_1} ) using two points (( x_1, y_1 )) and (( x_2, y_2 )).

• Example: Find the slope connecting points (4, 7) and (6, 10):

  • ( m = \frac{10 - 7}{6 - 4} = \frac{3}{2} )

• Point-Slope Form: If given a point ( (x_1, y_1) ) and slope ( m ):

  • The equation can be expressed as ( y - y_1 = m(x - x_1) ).

Types of Lines

• Horizontal Lines: These have the form ( y = k ) where ( k ) is a constant.

• Vertical Lines: These have the form ( x = k ) where ( k ) is a constant, and they are not functions.

Demand Function

• Definition: The price-demand function relates consumer demand for a product with its price, typically concluding that higher prices lead to lower demand.

• Linear Relationship: Generally modeled as a linear function with a negative slope:

  • ( p = mx + b ) (where ( m < 0 )).

• Example: For ( p = -0.2q + 10 ):

  • The slope of ( -0.2 ) indicates that for every $0.20 increase in price, demand decreases by one unit.

Supply Function

• Definition: The supply function relates the supply of a product to its price.

• Assumption: Generally modeled as a linear function with a positive slope:

  • ( p = mx + b ) (where ( m > 0 )).

• Example: For ( p = 0.2q + 5 ):

  • Indicates that for every $0.20 increase in price, the supply increases by one unit.

Functions and Their Properties

• General Definition: A function ( f ) from set ( X ) to set ( Y ) assigns exactly one element in ( Y ) to each element in ( X ).

• Notation: Written as ( y = f(x) ):

  • ( x ): input variable (independent)

  • ( y ): output variable (dependent).

Evaluating Functions

• Example: For ( f(x) = x^2 - 2 ):

  • To evaluate:

    • ( f(0) = -2 )

    • ( f(-2) = 2 )

    • ( f(3) = 7 )

    • Assume variables: ( f(h) = h^2 - 2 ) and ( f(x + 1) = (x + 1)^2 - 2 = x^2 + 2x - 1 ).

• Domain Definition: The domain consists of all acceptable inputs.

  • Example: In ( f(x) = 3/x^2 - 4 ), the domain excludes ( x = 2 ) due to undefined scenarios (division by zero).

Special Examples of Functions

• Square Root Functions: ( f(x) = \sqrt{x-2} )

  • Domain determination focuses on values where ( x-2 \geq 0 ) \Rightarrow Domain ( [2, \infty) )

Problem Solving with Demand and Supply Functions

• Example (Demand): Given ( p = 120 - \frac{2}{3}x ):

  • The slope ( -\frac{2}{3} ) indicates a decrease in demand with price increases.

  • The p-intercept ( 120 ) indicates maximum price at zero demand.

  • Determine quantity when price is $10, and graph demand function for ( 0 < x < 100 ).

• Example (Supply): Given ( p = 120 + \frac{1}{2}x ):

  • The slope ( \frac{1}{2} ) signifies an increase in supply with price increases.

  • The p-intercept is 120, interpreted similarly as demand.

  • Find supply for specified prices and graph for specified ranges.