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).