Algebraic reasoning and modeling

Question

A video game business finds there is a linear relationship between the number of games,

n

,it can sell and the price,

p

, it can charge per game. In particular, historical data shows that

1000

gamess can be sold at a price of

$

43

, while

9000

gamess can be sold at a price of

$

11

. Give a linear equation in the form

f

(

n

)

=

m

n

+

b

that gives the price

p

they can charge for

n

games.

Answer:

f

(

n

)

=

Round the value of your slope to three decimal places. Be careful to use the proper variable and use the Preview button to check your syntax before you submit your answer.

Answer 1

1. First, convert verbal descriptions into mathematical language.

n= # of games

p= price per game

1000 games for $43 per game —> (1000, 43)

9000 games for $11 per game —> (9000, 11)

Goal: p= f(n)=mn+b

*Need to find m (slope) and b (y-intercept)

1. To find slope (m), use slope formula, substituting point 1: (x1, y1)=(1000,43) and point 2: (x2,y2)=(9000,11)

m=slope= (y2-y1)/(x2-x1)= (11-43)/(9000-1000)=-32/8000=-0.004

1. to find y-intercept (b), choose either point (let’s choose (1000,43)), plug in slope (m=-0.004) and x and y coordinates of your point into the general form of the equation: f(n)=mn+b, and solve for b

43=-0.004(1000)+b

43=-4+b

47=b

final answer:

p=f(n)=-0.004n+47

Answer 2

Use the slope formula to calculate the slope, then use either one of the coordinate pairs to determine the y-intercept.

f(n) = -(1/250)n + 47

or

f(n) = -0.004n + 47