Formulas For Linear Functions (C: 1.4)
In this lecture:
1. Alternate Formulas for Equations of Lines
2. Three Alternate Forms for the Equation of a line
3. A Budget Constraint Application
4. Finding Formulas for Linear Functions From Graphs
________________________________________________
Alternate Formulas for Equations of Lines
Class Activity
(A) Graph the linear relationship 6x + 3y = 12 on the axes
below.
(B) Solve the relationship 6x + 3y = 12 for y in terms of x.
C 1.4 Survival Guide Notes copyright © 1998, 2007 Knobel/Stanley
1
Remark
Note that the graph of y = f (x) = !2x + 4 is exactly the same
as the graph of the relationship 6x + 3y = 12; that is, the
values of x and y that satisfy 6x + 3y = 12 are exactly the
same as the ordered pairs that satisfy the equation
y = –2x + 4.
(C) Solve the equation y ! 2 = –2(x –1) for y in terms of x.
Remark
Note that the same ordered pairs satisfy the three equations
6x+3y=12, y ! 2 = –2(x –1) and y = –2x + 4. The equations
represent the same line.

C 1.4 Survival Guide Notes copyright © 1998, 2007 Knobel/Stanley
2
Three Alternate Forms for the Equation of a Line
The three representations of lines we studied in the exercise
above are used in various ways in the work that follows. We
give them names here:
Description
Point-slope form
Slope-intercept form
General form
Equation
y ! y1 = m(x ! x1)
y = mx + b
Ax + By = C
Notes
(x1,y1) is ANY point
on the line and m
is the slope.
(0, b) is the yintercept of the line
and m is the slope.
A, B, and C are any
real numbers (but
A and B may not
both be zero).
Note: A given line
has many “general
forms.” For
example, x + y =1
and 3x + 3y = 3 are
different general
forms of the same
line.
C 1.4 Survival Guide Notes copyright © 1998, 2007 Knobel/Stanley
3
A Budget Constraint Application
Class Activity
Suppose you have a budget of $24 to spend on chips and
soda for a party. A six pack of soda costs $3 and a bag of
chips costs $1.50. Let x be the number of six packs
purchased and y be the number of bags of chips purchased.
(A) Write a linear relationship (in standard form) that relates
x and y.
(B) Find the x and y intercepts and label them on the graph
below.
C 1.4 Survival Guide Notes copyright © 1998, 2007 Knobel/Stanley
4
(C) Explain the practical significance of the x and y
intercepts.
(D) Express y, the number of bags of chips that can be
purchased as a function of x, the number of six packs that
can be purchased.
(E) What is the significance of the slope of the line?

Remark
Note that in the last exercise, both the standard form and the
slope intercept form for a linear equation provided us with
useful ways to think about the given situation. Which one is
used depends upon the focus of the problem.
C 1.4 Survival Guide Notes copyright © 1998, 2007 Knobel/Stanley
5
Finding Formulas for Linear Functions From Graphs
Class Activity
In a certain university you pay a membership fee and then all
of the meals are charged at some fixed price per meal. The
graph below shows that 30 meals cost $152.50 and 60
meals cost $250.
C (Cost in $)
• (60, 250)
•
(30, 152.50)
n (number of meals)
30
60
(A) Find C the total cost of the meal plan as a function of n
the number of meals, C = f (n).
C 1.4 Survival Guide Notes copyright © 1998, 2007 Knobel/Stanley
6
(B) Identify the fixed price per meal and the membership fee
in the formula for the function in part (A).
(C) Find n in terms of C.
(D) Find the maximum number of meals you can purchase
on a budget of $350

C 1.4 Survival Guide Notes copyright © 1998, 2007 Knobel/Stanley
7
C 1.4 Survival Guide Notes copyright © 1998, 2007 Knobel/Stanley
8