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Quadratic Functions

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 x-coordinate is -b/2a. Its y-coordinate is
given by

x- Intercepts (if any):

x-Intercepts (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 x-intercepts
are given by

If the discriminant is positive, there are two x-intercepts. If it is zero, there is a
single x-intercept (at the vertex). If it is negative , there are no x-intercepts (so the
parabola doesn ’t touch the x-axis at all).

y-Intercept:
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 log-ons per month. When
you lowered the price to $1.50, the demand increased to 560 log-ons per month.

a. Construct a linear demand function for your website and hence obtain the monthly
revenue R as a function of the log-on 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 log-on fee x, and hence determine the log-on
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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