TYPES OF FUNCTIONS
a. Constant Functions f(x) = a b. Linear Functions f(x) = a1x + a0 c. Quadratic Functions f(x) = a2x2 + a1x + a0 d. Polynomial Functions f(x) = anxn + an-1xn-1 + …+ a1x + a0 e. Rational Functions g ( x) f(x) = h( x )
APPLICATION FUNCTIONS
a. Linear Demand Functions, p = f(q) p = price; q = quantity of a product
b. Linear Supply Functions, p = f(q) p = price; q = quantity of a product
c. Linear Cost Functions, C(q) = fixed cost + variable cost q = number of units produced
d. Linear Revenue Functions, R(q) = p q q = number of units sold; p = price
e. Linear Profit Functions, π(q) = R(q) – C(q) q = quantity of a product
FORMING LINEAR EQUATIONS
I. Standard form of Linear equation Ax + By = C where A,B, and C are constants (A & B not both 0) ===> is a straight line
It is a first-degree equation – each variable in the equation is raised to the first power. e.g.: 2x + 5y = -5 2s – 4t = -1/2
FORMING LINEAR EQUATIONS
II. Slope-Intercept form of Linear equation y = mx + b
Example:
y = 2x + 1 y = -3x + 2
FORMING LINEAR EQUATIONS
III. Point-Slope form of Linear equation y – y1 = m (x – x1)
Is an equation of a line with slope m that passes through (x1, y1)
It enables us to find an equation for a line if given
a) its slope and the coordinates of a point b) two coordinates of two points on the line
SLOPE
-Slope
of two points (x1,y1) & (x2, y2) is given by
y2 y1 m x2 x1 y where x2 x1
(x2,y2)
Slope = Rise/Run Rise (y2-y1) x (x1,y1)
Run (x2-x1)
(x2,y1)
Example 1
Graph 3x – 4y = 12
Example 2
Find the slope of the line through each pair of points: a) (-2,5), (4,-7) b) (-3,-1), (-3,5)
Solution
Example 3
Find an equation for the line that has slope -2/3 and passes through (6,-1).
Solution
Example 4
Find an equation for the line that passes through the two points (1,-3) and (4, 3).
Solution
Example 5
A company that manufactures shoes has fixed costs of $250 per day and total cost of $3,450 at an output of 80 pairs of shoes per day. Assume that cost C is linearly related to output q. a) Find the slope of the line joining the points associated with outputs of 0 and 80; b) Find an equation of the line relating output to cost. Write the final answer in the form C = mq + b.
Solution
Solution
Example 6
For every car service, the mechanic, Aziz, charges RM 60 for labour and an additional RM 20 per hour. How much will the customer be charged if the mechanic took 3.25 hours to service his car?
Example 7
Ah Chong who is Aziz’z competitor has a car service shop nearby. He charges RM 45 for labour and an additional RM35 per hour. a. How much will the customer pay if Ah Chong took 3 hours to service his car? b. Who charges lesser for the car service: Aziz or Ah Chong?
Example 8
Mr. Oz was driving from KL to Penang at a constant rate of speed. After 2 hours, he noticed that he was 210 km away and later after 1 hour, he realized that he is 100 km away. a. Write a linear equation to represent the distance Mr. Oz is away, D as a function of time (t). b. Draw the linear equation. c. Interpret the y-intercept and the slope of the equation.
