L11 QUADRATIC FUNCTIONS
A quadratic function is a function of the form
f (x) = ax2 + bx + c
where a, b and c are real numbers and a = 0. The domain of a
quadratic function is the set of all real numbers.
a = 1 and b = c = 0
a = −1 and b = c = 0
f (x) = x2 − 4x + 5
The equation of a quadratic function
f (x) = ax2 + bx + c
a = 0, can be written in the form
f (x) = a(x − h)2 + k
h=− and k = f (h)
The graph of the equation y = a(x − h)2 + k for a = 0 is
a parabola that has vertex V (h, k) and a vertical axis. The
parabola is oriented upward if a > 0 and is oriented down-
ward if a < 0.
Express in the form f (x) = a(x − h)2 + k and sketch the graph
of the following:
f (x) = 3x2 + 24x + 50
f (x) = −x2 + 6x + 3
Find the vertex, intercepts, domain and range. Graph the
f (x) = 2x2 + 12x + 10
Find the equation of the quadratic function that has the vertex
V (−2, −3) and whose graph passes through the point P (−1, 0)
Modeling with quadratic functions
Let f (x) = a(x − h)2 + k, with a = 0
If a > 0 then f has a minimum value f (h) = k
If a < 0 then f has a maximum value f (h) = k
Suppose that a baseball is tossed straight up, and its height s
(in feet) as a function of time t (in seconds) is given by
s(t) = −16t2 + 64t + 6
where t = 0 corresponds to the time when the ball is released.
When does the ball reach the maximum height?
What is the maximum height of the ball?
A builder has 800 feet of fence left over from a job. He wants
to fence in a rectangular plot of land except for a 20-foot strip
to be used as a driveway.
Express A the are of the plot as a function of x
For what x is the area A a maximum?
What is the maximum area that he can enclose?
Solve: x2 − 5x + 6 ≥ 0
−x2 + 3x + 4 < 0
Break-Even, Proﬁt or Loss
The price p in dollars of selling x gallons of paint is given by
p(x) = 100 − .1x
The cost of producing x-gallons of paint is given by
C(x) = 10x + 8000
The revenue is R = xp and the company breaks even if
the cost equal revenue. The company makes a proﬁt if the
R > C. Find the break-even points. For what values of x does
the company make a proﬁt?