Parabola Problem with Solution (1)
A problem involving a parabola and a line is presented along with detailed solution.
Problem:
The graphs of a parabola given by y = f(x) and that of a line given by y = g(x) are shown below. The parabola and the line intersect at points A and D; where A is on the xaxis and has an xcoordinate equal to 2. V is the vertex of the parabola.
.
a) Find the equation of g and write it of the form g(x) = m x + b
b) Find the equation of the parabola.
c) Find x coordinate of point D.
d) Use the graph to solve the inequality f(x) > g(x).
Solution to Problem 1:

a) y = g(x) is the equation of a line through the points C(3,5) and A(2,0)
slope = (0(5)) / (2 (3)) = 1
equation of line: y  0 = 1(x  2) , which may be written as: y = x  2
hence g(x) = x  2

b) The vertex of the parabola is known V(4,4). The equation of the parabola in vertex form is given by
y = a(x  4)^{2}  4
Point A(2,0) is on the parabola hence: 0 = a(0  4)^{2}  4
solve the above equation for a to find a = 1
hence the equation of the parabola is given by: y = (x  4)^{2}  4

c) The x coordinate of point D is determined by solving the equation
(x  4)^{2}  4 = x  2 (intersection of line and the parabola)
solve the above equation to obtain the solutions x = 2 and x = 7.
x = 2 is already known, it is the x coordinate of A. x = 7 is the x coordinate of point D.
 d) From the left side of the graph to point A, f(x) > g(x) and therefore the interval (infinity , 2) is a set of solutions. Also from point D to the right side of the graph f(x) > g(x) and therefore another set of solutions is given by the interval (7 , + infinity).
The solution set of the inequality f(x) > g(x) is given by:
( infinity , 2) U (7 , + infinity)
More math problems with detailed solutions in this site.


Home Page 
HTML5 Math Applets for Mobile Learning 
Math Formulas for Mobile Learning 
Algebra Questions  Math Worksheets

Free Compass Math tests Practice
Free Practice for SAT, ACT Math tests

GRE practice

GMAT practice
Precalculus Tutorials 
Precalculus Questions and Problems

Precalculus Applets 
Equations, Systems and Inequalities

Online Calculators 
Graphing 
Trigonometry 
Trigonometry Worsheets

Geometry Tutorials 
Geometry Calculators 
Geometry Worksheets

Calculus Tutorials 
Calculus Questions 
Calculus Worksheets

Applied Math 
Antennas 
Math Software 
Elementary Statistics
High School Math 
Middle School Math 
Primary Math
Math Videos From Analyzemath
Author 
email
Updated: 2 April 2013
Copyright © 2003  2014  All rights reserved