Review: Completing the Square

Section: 
Review for 7.4
Date: 
Thursday, September 2, 2010 - 14:00
AttachmentSize
fall2010math2300_completing_square.pdf66.83 KB

Math 2300 Section 005 – Calculus II – Fall 2010

Completing the Square Examples – September 2, 2010

  1. Put the quadratic f(x)=x^{2}-2x+5 in vertex form. Explain the transformations required to translate g(x)=x^{2} into f(x), allowing the graph of f(x) to be given.

    Solution:

    \displaystyle f(x) \displaystyle=x^{2}-2x+5
    \displaystyle f(x) \displaystyle=x^{2}-2x+1-1+5
    \displaystyle f(x) \displaystyle=(x^{2}-2x+1)+4
    \displaystyle f(x) \displaystyle=(x-1)^{2}+4.

    Starting with g(x)=x^{2}, we first shift one unit right and obtain (x-1)^{2}, then shift four units up and obtain (x-1)^{2}+4.

  2. Find the center and radius of the circle

    x^{2}-6x+y^{2}+8y-20=2.

    Solution: By completing the square on each variable, we have (notice that this time we are adding the required constants on both sides of the equation, instead of adding and subtracting on the same side)

    \displaystyle x^{2}-6x+y^{2}+8y-20 \displaystyle=2
    \displaystyle x^{2}-6x+9\quad+\quad y^{2}+8y+16 \displaystyle=2+20+16
    \displaystyle(x-3)^{2}+(y+4)^{2} \displaystyle=38.

    The center is thus (3,-4) and the radius is r=\sqrt{38}.

© 2011 Jason B. Hill. All Rights Reserved.