# 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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