SECTION 3.1GRAPHING QUADRATICFUNCTIONS IN APQ FORM:
REVIEW: LINEAR AND QUADRATIC FUNCTIONS
Linear Functions
Straight Lines
General Form:
Highest degree for “x” is one
Quadratic Functions
Curved
Shape of a “Parabola”
Highest Degree for “x” is two
Vertex Form: (APQ)
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I) WHY IS A QUADRATIC FUNCTION U-SHAPED?
If we make a TOV, plot the coordinates, andconnect the dots, the resulting shape is a Parabola
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x
y
0
II) COMPONENTS OF A PARABOLA
Vertex: The tip of theparabola
Axis of Symmetry: A linethat cuts the graph inhalf (middle)
X intercepts: intersectionpoint between graph andthe x axis
Y intercept: intersectionpoint between graph andthe y axis
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III) GRAPHING QF:
A Quadratic function in vertex form is much easier tograph
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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IV) HORIZONTAL TRANSLATIONS
A parabola will shift left or right depending on whatconstant you place inside the brackets with “x”
Draw a circle around the “x”only, ignore any exponentssquares, radicals, ….
x
y
-5
-4
-3
-2
-1
0
1
2
3
4
5
-1
1
2
3
4
5
6
7
8
Graph is shifted
3 units to the right
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x
y
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
-1
1
2
3
4
5
6
7
8
EX: GIVEN THE FOLLOWING EQUATIONSA) GRAPH BOTH EQUATIONS,B) INDICATE WHAT TRANSLATIONS OCCURRED
x
y
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
-1
1
2
3
4
5
6
7
8
Graph is shifted
4 units to the LEFT
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III) VERTICAL TRANSLATIONS (VT)
A Vertical shift (UP or Down) will occur if a constantis added to the equation outside of x2
The value of “x” is squared first andthen we subtract the 3
Graph is shifted
3 units Down
The constant “3” is subtracted outsideof the brackets
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x
y
-5
-4
-3
-2
-1
0
1
2
3
4
5
-1
1
2
3
4
5
6
7
8
EX: GIVEN THE FOLLOWING EQUATIONSA) GRAPH BOTH EQUATIONS,B) INDICATE WHAT TRANSLATIONS OCCURRED
Graph is shifted 2 units UP
x
y
-5
-4
-3
-2
-1
0
1
2
3
4
5
-1
1
2
3
4
5
6
7
8
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IV) SUMMARY FOR 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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EX: FOR EACH OF FOLLOWING EQUATIONS, FIND THECONSTANTS “A”, “P”, “Q”, VERTEX, AND A.O.S.
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V) 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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VI) 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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HOMEWORK
Assignment 3.1
x
y
0