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Quadratic Functions
A quadratic function has the form f(x) = ax^2 + bx + c (a
≠ 0). The graph of a quadratic
function is a parabola. If a > 0 the parabola opens upward (concave up)
and if a < 0 it
opens downward (concave down)
Vertex:
The vertex is the turning point of the parabola. Its
xcoordinate is b/2a. Its ycoordinate is
given by
x Intercepts (if any):
xIntercepts (if any):
These occur when f (x) = 0; that is, when
Solve this equation for x by either factoring or using the
quadratic formula. The xintercepts
are given by
If the discriminant
is positive, there are two xintercepts. If it is zero, there is a
single xintercept (at the vertex). If it is negative , there are no xintercepts
(so the
parabola doesn ’t touch the xaxis at all).
yIntercept:
This occurs when x = 0, so y = a(0)2 + b(0) + c = c
Symmetry:
The parabola is symmetric with respect to the vertical line through the
vertex, which is the line
Quadratic Regression Curve:
In Section 1.5 we saw how to fit a regression line to a collection of data
points. Here, we
use technology to obtain the quadratic regression curve associated with a
set of points.
The quadratic regression curve is the quadratic curve
that best fits the
data points in the sense that the associated sum of squares error (SSE—see
Section 1.5)
is a minimum. Although there are algebraic methods for obtaining the quadratic
regression curve, it is normal to use technology to do this.
Problem 1. The Better Baby Buggy Co. has just come
out with a new model, the Turbo.
The market research department predicts that the demand equation for Turbos is
given by
q = −2p + 320, where q is the number of buggies it can sell in a month if the
price is $p
per buggy. At what price should it sell the buggies to get the largest revenue?
What is the
largest monthly revenue?
Problem 2. The average weight of an SUV could be
approximated by
where t is its year of manufacture (t = 0
represents
1970) and W is the average weight of an SUV in pounds. Sketch the graph of W as
a
function of t. According to the model, in what year were SUVs the lightest? What
was
their average weight in that year?
Problem 3. You operate a gaming website, where
users must pay a
small fee to log on . When you charged $2 the demand was 280 logons per month.
When
you lowered the price to $1.50, the demand increased to 560 logons per month.
a. Construct a linear demand function for your website and
hence obtain the monthly
revenue R as a function of the logon fee x.
b. Your Internet provider charges you a monthly fee of $30
to maintain your site. Express
your monthly profit P as a function of the logon fee x, and hence determine the
logon
fee you should charge to obtain the largest possible monthly profit. What is the
largest
possible monthly profit?
Problem 4. The following table shows the value of
U.S. trade with China in 1994, 1999,
and 2004 (t = 0 represents 1994).
Year t  0  5  10 
China Trade ($ Billion)  50  95  275 
Find a quadratic model for these data, and use your model
to estimate the value of U .S.
trade with China in 2000.
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