English | Español

Try our Free Online Math Solver!

Online Math Solver












Please use this form if you would like
to have this math solver on your website,
free of charge.

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)


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).

This occurs when x = 0, so y = a(0)2 + b(0) + c = c

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.

Prev Next