f(x) = ax2 + bx + c
the graph of a quadratic function of the form
has a vertex at the point (h , k) where h and k are given by
h = -b/2a
k = c - b2/4a
also, k = f(h).
If a > 0, the vertex is a minimum point and the minimum value of the quadratic function f is equal to k. This minimum value occurs at x = h = -b/2a.
If a < 0, the vertex is a maximum point and the maximum value of the quadratic function f is equal to k. This maximum value occurs at x = h = -b/2a.
Example - Problem 1 : The profit (in thousands of dollars) of a company is given by.
P(x) = 5000 + 1000x - 5x2
where x is the amount ( in thousands of dollars) the company spends on advertising.
- Find the amount, x, that the company has to spend to maximize its profit.
- Find the maximum profit Pmax.
Solution to Problem 1:
- Function P that gives the profit is a quadratic function with the leading coefficient a = -5. This function (profit) has a maximum value at x = h = -b/2a
x = h = -1000/2(-5) = 100
- The maximum profit Pmax, when x = 100 thousands is spent on advertising, is given by the maximum value of function P
k = c - b2/4a
- The maximum profit Pmax, when x = 100 thousands is spent on advertising, is also given by P(h = 100)
P(100) = 5000 + 1000(100) - 5(100)2 = 55000.
- When the company spends 100 thousands dollars on advertising, the profit is maximum and equals 55000 dollars.
- Shown below is the graph of P(x), notice the maximum point, vertex, at (100 , 55000).
Example - Problem 2 : An object is thrown vertically upward with an initial velocity of Vo feet/sec. Its distance S(t), in feet, above ground is given by
S(t) = -16t2 + vot.
Find vo so that the highest point the object can reach is 300 feet above ground.
Solution to Problem 2:
- S(t) is a quadratic function and the maximum value of S(t)is given by
k = c - b2/4a = 0 - (vo)2 / 4(-16)
- This maximum value of S(t) has to be 300 feet in order for the object to reach a maximum distance above ground of 300 feet.
- (vo)2 / 4(-16) = 300
- we now solve - (vo)2 / 4(-16) = 300 for vo
vo = 64*300 = 80sqrt(3) feet/sec.
The graph of S(t) for vo = 64*300 = 80sqrt(3) feet/sec is shown below.
More references and links on the quadratic functions in this website.