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:

We'll do this any time we encounter a linear function.

If the price is 30 then we get

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.

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.

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 we sell demand(30) = 110 units, as we found before.

Our revenue will therefore be

revenue = price * number sold = $30 * 110 = $3300.

The demand will be demand(p) = 200 - 3 * p.

Revenue will be

revenue = price * number sold = price * demand = p * ( 200 - 3 p).

The revenue function will be

revenue(price) = price * demand =

price * (200 - 3 * price).

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

We find the following:

So the vertex is at (33.33, 3333).

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 graph is a parabola opening downward, with vertex (33.33, 3333) and zeros at (0, 0) and (66.67, 0).