Quadratic Functions Problems with Solutions

quadratic functions problems with detailed solutions are presented along with graphical interpretations of the solutions.

Review Vertex and Discriminant of Quadratic Functions

the graph of a quadratic function written in the form

f(x) = a x 2 + b x + c

has a vertex at the point (h , k) where h and k are given by
h = - b / (2 a)   and   k = f(h) = c - b 2 / (4 a)

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.
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.
The quadratic function f(x) = a x 2 + b x + c can be written in vertex form as follows:

f(x) = a (x - h) 2 + k



The discriminant D of the quadratic equation: a x 2 + b x + c = 0 is given by D = b2 - 4 a c
If D = 0 , the quadratic equation a x 2 + b x + c = 0 has one solution and the graph of f(x) = a x 2 + b x + c has ONE x-intercept.
If D > 0 , the quadratic equation a x 2 + b x + c = 0 has two real solutions and the graph of f(x) = a x 2 + b x + c TWO two x-intercepts.
If D > 0 , the quadratic equation a x 2 + b x + c = 0 has two complex solutions and the graph of f(x) = a x 2 + b x + c has NO x-intercept.

Problems with Solutions

Problem 1
The profit (in thousands of dollars) of a company is given by.

P(x) = 5000 + 1000 x - 5 x2

where x is the amount ( in thousands of dollars) the company spends on advertising.
a) Find the amount, x, that the company has to spend to maximize its profit.
b) Find the maximum profit Pmax.
Solution to Problem 1

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) = -16 t 2 + vo t.

Find vo so that the highest point the object can reach is 300 feet above ground.
Solution to Problem 2

Problem 3
Find the equation of the quadratic function f whose graph passes through the point (2 , -8) and has x intercepts at (1 , 0) and (-2 , 0).
Solution to Problem 3


Problem 4
Find values of the parameter m so that the graph of the quadratic function f given by

f(x) = x 2 + x + 1
and the graph of the line whose equation is given by
y = m x

have:
a) 2 points of intersection,
b) 1 point of intersection,
c) no points of intersection.
Solution to Problem 4

Problem 5
The quadratic function C(x) = a x 2 + b x + c represents the cost, in thousands of Dollars, of producing x items. C(x) has a minimum value of 120 thousands for x = 2000 and the fixed cost is equal to 200 thousands. Find the coefficients a,b and c.
Solution to Problem 5

Problem 6
Find the equation of the tangent line to the the graph of f(x) = - x 2 + x - 2 at x = 1.
Solution to Problem 6

Questions with Solutions

Question 1

Find the equation of the quadratic function f whose graph has x intercepts at (-1 , 0) and (3 , 0) and a y intercept at (0 , -4).

Question 2
Find values of the parameter c so that the graphs of the quadratic function f given by
f(x) = x 2 + x + c
and the graph of the line whose equation is given by y = 2 x
have:
a) 2 points of intersection,
b) 1 point of intersection,
c) no points of intersection.

Solutions to the Above Questions

Solution to Question 1

Solution to Question 2


More References and Links to Quadratic Functions