MPM2D U05L02
Converting Standard Form to Vertex Form
Consider the vertex form of a quadratic relation:
y = a(x - h)
2
+ k
Vertex form is made up of ("a") groups of perfect squares with side length (
x - h)
and a constant term ("
k")
added. So, if we want to convert to vertex form, we
must create (or "complete") a perfect square first.
Recall: (x + __)
2
= x
2
+ 2(__)x + (__)
2
or (x - __)
2
= x
2
- 2(__)x + (__)
2
Example # 1: Write y = x
2
+ 4x + 7 in vertex form.
Vertex form requires a perfect square; focus on creating that square. Ignore the
constant term 7 for now.
Use an area diagram. Place the
x
2
first; in order for us to make a square, the
bx
term must be broken up evenly; half on each side.
Tocompletethesquare,weneed4unit
tiles.Todothis,weadd4zeropairs!
+
+
Tofinish,addthesevenunit
tilesweignoredinthe
beginningandsimplify!
Note:Itwillalwaysbe
positivetilesneededto
completethesquare!!
MPM2D U05L02
Example#3:
Writeinvertexform:y=2x
2
+12x+3
Whenthereisanumericalcoefficientofx
2
otherthan1itisour"a"valueor
stretchfactor.Wemustdividethex
2
andxtilesinto"a"equalgroupsto
workwith.Completethesquareinonegroup.Weknowthenthatwehave
"a"ofthesquares,aswellas"a"timesthenumberoftilesleftoverfrom
completingthesquares.
Example#2:
Writeinvertexform:y=x
2
6x1
Formalsolutionshouldlook
likethis;yourworkwiththe
tilesgoesonthesideas
roughwork!
Nextstep:
Completethe
square!
MPM2D U05L02
AlgebraicMethod:
Example # 4:
Change y = 5x
2
+ 20x + 2 into vertex form by completing the square.
Steps:
•
commonfactorthenumericalcoefficientthat
isinfrontofthex
2
termoutofthefirsttwo
termsonly.Ignoretheconstanttermatthe
end.
•
createaperfectsquaretrinomialinsidethe
bracketbyaddingandsubtractingthesquare
ofhalfthenumericalcoefficientinfrontofthe
xterm.
•
usingonlythefirstthreeterms,changethe
perfectsquaretrinomialtoabinomialsquared.
•
rememberthatthefourthterminsidethe
bracketmustbeaffectedbythecommon
factorthatyouoriginallyremovedbeforeitcan
becollectedwithourconstanttermthatwe
havebeenignoring.
Example # 5:
Change y = -2x
2
+ 12x - 7 into vertex form by completing the square.
Homework:p.390#4,5,9ab