1. FORMULAS FOR LINEAR FUNCTIONS
1.4
FORMULAS FOR LINEAR FUNCTIONS

2. Finding a Formula for a Linear Function from a Table of Data
Example 1 The following table gives data from a linear function for a grapefruit thrown into the air. Find the formula for the function. Solution We will look for a function of the form y = mx + b and begin by computing the slope: Now we can determine b using m and the point (2, 16): y = -32x+ b; 16 = -32 (2) + b which gives b = 80 So the formula for our function is
t, time (sec) 1 2 3
v, velocity (ft/sec) 48 16
-16

3. Exercise 26 The following table gives data from a linear function
Exercise 26 The following table gives data from a linear function. Find a formula for the function. Solution We will look for a function of the form y = mx + b and begin by computing the slope: Now we can determine b using m and the point (32, 0): y = 5/9 x + b; 0 = 5/9 (32) + b which gives b = -160/9 So the formula for our function is
Temperature, x (◦F) 32 41 68
Temperature, y = f(x) (◦C) 5 20

4. Finding a Formula for a Linear Function from a Graph
Exercise 30
Solution
First we will find the slope of the line using (4, 7) and (12, 3).
We will look for a function of the form y – y0 = m (x – x0) using (4, 7)
y – 7 = – 0.5 (x – 4)
y – 7 = – 0.5x + 2
which simplifies to
y = – 0.5 x + 9 y
The graph gives data from a linear function. Find a formula for the function. x

5. Finding a Formula for a Linear Function from a Verbal Description
Example 3
We have $24 to spend on soda and chips for a party. A six-pack of soda costs $3 and a bag of chips costs $2. The number of six-packs we can afford, s, is a function of the number of bags of chips we decide to buy, c.
Find an equation relating s and c.
Solution
The amount of money ($) spent on soda will be 3s.
The amount of money ($) spent on chips will be 2c.
Assuming we spend all $24, the equation becomes:
2c + 3s = 24
3s = -2c + 24
s = – 2/3 c + 8

6. Interpreting a Formula for a Linear Function from a Verbal Description
Example 3
From (a), the equation is
(b) Graph the equation. Interpret the intercepts and the slope in the
context of the party.
Solution
All soda
No chips
The fact that m = −2/3 means that for each additional bag of chips purchased, we can purchase 2/3 fewer six-packs of soda.
6 packs of soda and 4 bags of chips
All chips
No soda

7. Alternative Forms for the Equation of a Line
The slope-intercept form is
y = mx + b
where m is the slope and b is the y-intercept.
The point-slope form is
y − y0 = m(x − x0)
where m is the slope and (x0, y0) is a point on the line.
The standard form is
Ax + By + C = 0
where A, B, and C are constants.