Download Vertical Intercept

Shown below is another demand function for price of a pizza p as a function of the quantity of pizzas sold per week. 30
25
price p
20
D (q ) = −.0016q 2 − .08q + 24
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quantity q
To insure that the shape of this demand function makes sense, let’s look at two different regions of the graph. On the left side of the graph, the price is high and the quantity is low. In fact, the price of the pizza is so high that almost no pizzas are sold. On the right side of the graph, the price is low and the quantity sold is higher. 30
Price p is high and the quantity q is low 25
price p
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Price p is low and the quantity q is high 5
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quantity q
In this example, I want to find and interpret what the intercepts mean. Vertical Intercept At the vertical intercept, the value of q is zero. To find the corresponding value of p, we need to use the function’s formula since it takes input of q and gives outputs of p, p = D(q ) . Substituting q = 0 into the function yields p = D(0) = −.0016 ( 0 ) − .08 ( 0 ) + 24 = 24 2
So the vertical intercept is at (24, 0). This means that a price of $24 per pizza, none are sold each week. The pizzas are so expensive that no customers buy them. Horizontal Intercept At the horizontal intercept, the value of p is equal to zero. To find the horizontal intercept, we need to find the value of q that corresponds to p = 0. You can do this by solving the equation −.0016q 2 − .08q + 24 = 0 This is a quadratic equation with a = ‐.0016, b = ‐.08 and c = 24. It might look a bit different because it is a quadratic equation in the variable q instead of the variable x. To solve this equation for q, we’ll use the quadratic formula q=
−b ± b 2 − 4ac
2a
If we substitute the values of a, b, and c into the quadratic formula we get q=
−(−.08) ± (−.08) 2 − 4(−.0016)(24)
2(−.0016)
If you are very careful with parentheses, you can calculate this expression on your calculator. The intercept at q = 100 matches the graph. The other intercept is in quadrant 2, but we’ll ignore it since quantities can’t be negative. The intercept at (100, 0) means that at a price of $0, the quantity sold is 100 pizzas. In other words, you can give away 100 pizzas per week.