0302
If demand(price) = 200 - 3 * price, where price is in dollars, then
Good idea to immediately graph the function.
We recognize that demand is a linear function of price, since we have just the first power of the price.
For any function it's a good practice to find the x and y intercepts.
In this case we find the y = demand intercept by letting x = price be zero. We get demand(0) = 200 - 3 * 0 = 200. So one intercept is (0, 200).
To get the x = price intercept let y = demand be zero, obtaining the equation
0 = 200 - 3 * price,
which we easily solve to get
price = 200/3 = 66.67 approx.
This gives us the intercept (200/3, 0) = (66.67, 0) approx..
We also look at the basic point 1 unit to the right of the y axis:
If we move 1 unit to the right, price increases by 1 unit. Since the price dependence is determined by -3 * price, we see that the demand must go down by 3.
Another way of looking at this is to see that the slope of the graph is -3. If y = demand and x = price, then we have y = 200 - 3 x, which can be written y = -3x + 200. This is slope-intercept form with slope -3 and y-intercept 200.
What we've done is:
Recognize that this is a linear function.
Find the intercepts.
Find the basic points.
Find the slope.
Sketch the graph.
We'll do this any time we encounter a linear function.
How many units will be demanded if price = 30?
If the price is 30 then we get
demand(30) = 200 - 3 * 30 = 110.
By how much does demand change for every additional dollar of price? Does the answer depend on the iniitial price?
If price changes by 1 then -3 * price changes by -3. This changes demand(price) = 200 - 3 * price by -3.
Every dollar of increase in price decreases demand by 3 units.
At what price will demand be zero?
Demand is zero if 0 = 200 - 3 * price. We already have this information on our graph, and we solved this equation to find the x = price intercept. The intercept is (66.67, 0).
This tells us that if we charge $66.67 we won't sell anything.
What is the demand if the commodity is given away free?
If we give the thing away then the price is zero, so the demand is demand(0) = 200 - 3 * 0 = 200.
Giving the product away for free corresponds to the y intercept of the graph.
If we set the price at $30 then how many will we sell and how much money will we get?
If we set the price at $30 then we sell demand(30) = 110 units, as we found before.
Our revenue will therefore be
revenue = price * number sold = $30 * 110 = $3300.
If we set the price at p then how many units will we sell and what will be our revenue?
The demand will be demand(p) = 200 - 3 * p.
Revenue will be
revenue = price * number sold = price * demand = p * ( 200 - 3 p).
Assuming that every object demanded is sold at the corresponding price, what function describes the number of dollars of revenue as a function of price?
The revenue function will be
revenue(price) = price * demand =
price * (200 - 3 * price).
What kind of function is this?
If we multiply the above expression out we get
revenue(price) = price * (200 - 3 * price) =
-3 * price^2 + 200 * price .
This is a quadratic function of the form y = a x^2 + b x + c with a = -3, b = 200, c = 0
Knee-jerk reaction to a quadratic function:
Analyze it by finding
vertex and basic points
zeros
intercepts
We find the following:
The vertex occurs when x = price = -b / (2a) = -200 / (2 * -3) = 33.33.
The y = demand intercept of the vertex is y = demand(33.33) = -3 * 33.33^2 + 200 * 33.33 = -3333 + 6667 = 3333.
So the vertex is at (33.33, 3333).
The zeros can be found by the quadratic formula. However since
demand(price) = price ( 200 - 3 * price)
we see that demand will be zero when price = 0 and also when 200 - 3 * price = 0.
Thus demand is zero if price = 0 or if price = 66.67.
The quadratic formula with a = -3, b = 200, c = 0 would give us the same result.
The y = demand intercept is (0, 0).
The points 1 unit right and left of the axis of symmetry lie -3 units from that axis, since a = -3 is the initial vertical stretch of the basic quadratic function used to obtain the present function. So these points are (32.33, 3330) and (34.33, 3336).
Describe the graph of revenue vs. price.
The graph is a parabola opening downward, with vertex (33.33, 3333) and zeros at (0, 0) and (66.67, 0).