Quadratic Functions - Problems (1)

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.

1. Find the amount, x, that the company has to spend to maximize its profit.
   
   
2. Find the maximum profit Pmax.

Solution to Problem 1:

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
   
   
2. 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
   
   
3. 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.
   
   
4. When the company spends 100 thousands dollars on advertising, the profit is maximum and equals 55000 dollars.
   
5. 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 V_o feet/sec. Its distance S(t), in feet, above ground is given by

Solution to Problem 2:

1. S(t) is a quadratic function and the maximum value of S(t)is given by
   k = c - b²/4a = 0 - (v_o)² / 4(-16)
   
   
2. 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.
   - (v_o)² / 4(-16) = 300
   
   
3. we now solve - (v_o)² / 4(-16) = 300 for v_o
   v_o = 64*300 = 80sqrt(3) feet/sec.

The graph of S(t) for v_o = 64*300 = 80sqrt(3) feet/sec is shown below.

• Quadratic functions (general form).
  
  
• Tutorial on Quadratic Functions (1).
  
  
• Quadratic functions (standard form).
  
  
• Graphing quadratic functions.