This is a tutorial on quadratic functions. The solutions and the explanations are detailed.
Example 1 : 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 Example 1:

Because the graph has x intercepts at (1 , 0) and (2 , 0), the equation of the function may be written as follows.
f(x) = a(x  1)(x + 2)

The graph of f passes through the point (2 , 8), it follows that
f(2) = 8

which leads to
8 = a(2  1)(2 + 2)

expand the right side of the above equation and group like terms
8 = 4a

Solve the above equation for a to obtain
a = 2

The equation of f is given by
f(x) = 2(x  1)(x + 2)

Check answer
f(1) = 0
f(2) = 0
f(2) = 2(2  1)(2 + 2) = 8
Matched Exercise: 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).
Answers to above exercise.
Example 2 : 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 = mx
have:
a) 2 points of intersection,
b) 1 point of intersection,
c) no points of intersection.
Solution to Example 2:

To find the points of intersection, you need to solve the system of equations
y = x^{2} + x + 1
y = mx

Substitute mx for y in the first equation to obtain
mx = x^{2} + x + 1

Write the above quadratic equation in standard form.
x^{2} + x(1  m) + 1 = 0

Find the discrimant D of the above equation.
D = (1m)^{2}  4(1)(1)
D = (1m)^{2}  4

For the graph of f and that of the line to have 2 points of intersection, D must be positive, which leads to
(1m)^{2}  4 > 0

Solve the above inequality to obtain solution set for m in the intervals
(infinity , 1) U (3 , +infinity)

For the graph of f and that of the line to have 1 point of intersection, D must be zero, which leads to
(1m)^{2}  4 = 0

Solve the above equation to obtain 2 solutions for m.
m = 1
m = 3

For the graph of f and that of the line to have no points of intersection, D must be negative, which leads to
(1m)^{2}  4 < 0

Solve the above inequality to obtain solution set for m in the interval
(1 , 3)
The graphs of y = 3x, y = x and that of the quadratic function are shown in the figure below.
Matched Exercise: Find values of the parameter c so that the graph 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 = 2x
have:
a) 2 points of intersection,
b) 1 point of intersection,
c) no points of intersection.
Answers to above exercise.
More references and links on the quadratic functions in this website.
