SECTION 1.2QUADRATIC FUNCTIONS
i) Completing the square
ii) Graphing y=a(x-p)2+q
iii) Using a,p,q to find vertex, opens up, down,congruency factor,
iv) Deriving and using the quadratic function
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I) GRAPHING QF:
A Quadratic function in standard form is much easierto graph
Using constants “a”,”p”, & “q”, we can find the vertex,which way it opens and the congruency value
Vertex: Axis of Symmetry:
Domain:Range:
Y intercept: make x=0, solve for y
X-intercept: make y=0, solve for x
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EX: FOR EACH OF FOLLOWING EQUATIONS, FIND THECONSTANTS “A”, “P”, “Q”, VERTEX, AND A.O.S.
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II) CONSTANTS “P” AND “Q”
The constant “p” affects the graph horizontally
When p=0, the graph is centered on the Y-axis
x
y
0
x
y
0
x
y
0
2 unitsRight
2 unitsLeft
x
y
0
x
y
0
2 unitsup
2 unitsDown
InteractiveApplet
The constant “q” affects the graph vertically
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III) HOW DOES THE CONSTANT “A” WORK?
5
5
3
3
1
1
Beginning at the vertexwe can graph all theother points withoutmaking a TOV
Each point increaseshorizontally by 1but increases verticallyby 1 , 3 , 5 , 7 , 9, …
7
7
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6
10
10
6
2
2
If “a = 2”, the pointsgo up faster.
Each point increaseshorizontally by 1but increases verticallyby 2 , 6 , 10 , 14 , 18, …
Simply multiply thevalues by “2”
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IV) CONSTANT “A” (CONGRUENCY FACTOR)
The constant “a” determines the (congruency) width ofthe parabola and which way it opens
If “a” is positive (Opens up)
If ‘a” is negative (Opens down)
If “a” is big (Skinny)
If “a” is small (Wide)
Congruency Factor:
The constant “a” can be used to determine how fastthe points on the parabola go up by
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PRACTICE: GRAPH THE FOLLOWING PARABOLAS ANDINDICATE THE VERTEX, AOS, DOMAIN & RANGE
2
6
10
3.5
2.5
1.5
0.5
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VI) CTS: COMPLETE THE SQUARE!
Completing the Square is a process that changes aquadratic functions from
General form: Standard Form:
Bracket the first two terms!
Divide the second term by 2 andsquare it! The purpose is to make theexpression in the bracket into aperfect square!
Take the negative square outside ofthe brackets!
The trinomial becomes two equalbinomials
Standard Form!
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PRACTICE: CONVERT THE FOLLOWING TO STANDARD FORM
Bracket the first two terms!
Divide the second term by 2 andsquare it!
Take the negative square outside ofthe brackets and multiply withcoefficient in front!
The trinomial becomes two equalbinomials
Standard Form!
Factor out any coefficient for x2
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HOMEWORK: ASSIGNMENT 1.2
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