Interpreting Key Features

Question

Jada sells ground paprika. Her weekly profit (in dollars) as a function of the amount of paprika she sold that week (in kilograms) is graphed. 

Write a function that models the situation.  

What is the domain of the function? What does it represent in the context of the problem?

What is the rate of change? What does it represent in the context of the problem?

What is the x-intercept? What does it represent in the context of the problem?

What is the y-intercept? What does it represent in the context of the problem? 

Over which interval, if any, is the function positive? What does it represent in the context of the problem? 

Over which interval, if any, is the function negative? What does it represent in the context of the problem?  

Answer 1

You are given TWO points (ordered pairs) of the line; (35, 0) and (60, 200)

You can use the POINT-SLOPE equation, (y - y₁) = m (x - x₁) to write a function that models the situation, using one of the points given (above), BUT you need to calculate the slope, m for it.

m = (y₂ - y₁)/(x₂ - x₁) = (200 - 0)/(60 - 35) OR (0 - 200)/(35-60) BOTH equal 8. m = 8

Use the x & y of ONE of the points and the slope calculated & plug into the POINT- SLOPE formula above.

Doesn't matter which point you use as P₁ or P₂...they will yield the same equation (see below.)

y - 0 = 8(x - 35)

y = 8x - (8)(35)

y = 8x - 280 ----- this is your function

OR

y - 200 = 8(x - 60)

y - 200 = 8x - (8)(60)

y - 200 = 8x - 480

y = 8x - 480 + 200

y = 8x - 280

The DOMAIN is the value of the 1st coordinate of ALL the ordered points, {35, 60} BUT it could be

{0, infinity} The DOMAIN here represents the amount of paprika that can be sold (throughout the life of the business.)

The RANGE is the value of the 2nd coordinate of ALL the ordered points, {0, 200} BUT it could be {0, infinity} The RANGE here represents the amount of money that can be lost and made (throughout the life of the business.)

The RATE OF CHANGE is the slope of the line and it represents the rate in which money was coming in (to the business) as paprika was sold. I don't say PROFIT because at the onset paprika was being sold but money was being lost.

The X-INTERCEPT is at (35,0) and I interpret that as the BREAK EVEN POINT, the quantity of paprika sold where money is neither being lost or profit made.

The Y-INTERCEPT (it hard to tell the # but it looks like 0, 275) and I interpret that as the money lost when ZERO paprika was been brought.

My interpretation: The interval from which the function is positive is from 35 to infinite (kilograms of paprika sold.) Once the business sold 35 kilograms of paprika, they started making profits.

My interpretation: The interval for which the function is negative is 0 - 34 (kilograms of paprika) - 35 kilograms is the break even point. From 0 - 34 kilograms of sold paprika yielded no profit & maybe cost the business money so the function in negative in this interval.

Hope this helps...