Linear Function

Question

Daily sales of gasoline y at the gas and go are 500 gallons at a price of $2.50, but fall to 425 gallons at a price of $2.80. Let X the price of gasoline per gallon.

write linear function that models this data. Then use this to find the given values

Answer 1

Right up front, were told that this is a linear function. A linear function is a function where the ratio between the change in y and the change in x is always the same for any 2 points on the graph or any two (x,y) value sets. This means that

(y-y₀)/(x-x₀)=m, where "m" is a constant (never changes), (x₀,y₀) is any specific given value set, or point on the line, and (x,y) is any point, or set of values, other than (x₀,y₀).

(y-y₀)/(x-x₀)=m

Multiply both sides of the equation by (x-x₀)

y-y₀= m(x-x₀)

This is called point slope format of a linear equation and is a form you would typically memorize and jump to without the prior steps explaining how to get there. But understanding the steps makes it easier to know how to use and remember (or re-establish if we forget.

This is a good place to start determining actual values and plugging them in.

First, we have two points to determine m, the unchanging ratio between the change in y and the change in x.

(500-425)/(2.50-2.80)= -250.

Always remember to set up these change in y versus change in x values in the same direction, point 1 to point 2 for both x and y (or point 2 to point 1 for both x and y).. NEVER do EITHER point 1 to point 2 for y and point 2 to point 1 for x OR point 2 to point 1 for y and point 1 to point 2 for x.

Second we pick a specific (x₀,y₀). We have two given points to choose from, and it doesn't make any difference which one we choose, (x₀,y₀)=(2.50, 500) or (2.80,425). So, let's pick (2.50, 500) because they are easier numbers to work with.

y-y₀= m(x-x₀)

y-500= -250(x-2.50)

Distribute the -250.

y-500= -250x-(-250)2.50

Add 500 to both sides of the equation.

y= -250x-(-250)2.50+500

Simplify

y= -250x+625+500

y= -250x+1125

In the slope intercept form of linear equation, y=mx+b, our b=1125.

This means that, even if we gave away the gasoline, x=$0, we would not distribute more than 1125 gallons.

I like this approach because you start with an understanding or re-enforcement of what a linear relationship is, and then do not get involved with simultaneous equations; AND when one get's into calculus, it is the point slope form of linear equation that has most relevance.

y-y₀= m(x-x₀)

but in the form, adding y₀ to both sides of the equation.

y= m(x-x₀)+y₀

And in function format:

f(x)=m(x-x₀)+f(x₀)

Answer 2

There is another way to solve this problem, using the two equations, two unknown method.

Assuming X is the price of gas per Gallon and Y is the total sale per day, then the linear function is Y = m X+b, so we have:

500 = m(2.50) + b

425 = m (2.80) + b

Subtracting both sides will result: 75 = - .3 m and therefore: m = -250. by substituting m to one of the above equations we have: 500 = -250 (2.5) + b or 500 = -625 + b or b = 1125 so the required function is :

Y = -250m + 1125. You can plug any value for X or Y and find the other one.

Answer 3

The demand line is y = f(x) = mx + b, where x is the price and y is the # of gallons sold.

500 gallons are sold when the price is $2.50, so (2.50, 500) is one point on this demand line.

But when the price increases to $2.80, only 425 gallons are sold, so the second point on the

demand line is (2.80, 425)

THe slope is (425 - 500) / (2.80 - 2.50) = -75/ 0.3 = -750/3 = -250

So the # of gallons sold DECREASES by 250 for every penny the price increases

Finally the intercept is B = y - mx. Using (2.50, 500), the intercept is

B = 500 - (-250)(2.50) = 500 + 250*2.5 = 500 + 625 = 1125

the demand line is y = -250x + 1125

Answer 4

The demand line is y = f(x) = mx + b, where x is the price and y is the # of gallons sold.

500 gallons are sold when the price is $2.50, so (2.50, 500) is one point on this demand line.

But when the price increases to $2.80, only 425 gallons are sold, so the second point on the

demand line is (2.80, 425)

THe slope is (425 - 500) / (2.80 - 2.50) = -75/ 0.3 = -750/3 = -250

So the # of gallons sold DECREASES by 250 for every penny the price increases

Finally the intercept is B = y - mx. Using (2.50, 500), the intercept is

B = 500 - (-250)(2.50) = 500 + 250*2.5 = 500 + 625 = 1125

the demand line is y = -250x + 1125