FORMULAS FOR LINEAR FUNCTIONS 1.4 FORMULAS FOR LINEAR FUNCTIONS

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

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

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

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

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

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.