Graphing Parabolas

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Graphing Parabolas
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Pattern Probes

• Which of the following relationships describes
  the pattern 1,-1,-3,-5?
• A. tn -2n
• B. tn -2n 3
• C. tn 2n 3
• D. tn 2n

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Pattern Probes

• 2. What is the 62th term of the pattern
  1,9,17,25,?
• A. 489
• B. 496
• C. 503
• D. 207

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Quadratics Patterns

• Last time we learned that

Note Another phrase for linear pattern is an
arithmetic sequence.
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Lets Graph (Put on your graphing shoes and
graph in blue?)
Quadratics, as we know, do not increase by the
same amount for every increase in x. (How do we
know this?) Therefore, their graphs are not
straight lines, but curves. In fact, if youve
ever thrown a ball, watched a frog jump, or taken
Math 12, youve seen a quadratic graph!
Its table of values would be
y x2 y x2
x y
-3 9
-2 4
-1 1
0 0
1 1
2 4
3 9
Consider the function f(x) x2 - the simplest
quadratic (Why?)
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Graphing Cont
Domain xe R Range y e 0,8) or y 0
Axis of Symmetry x 0
x y
-3 9
-2 4
-1 1
0 0
1 1
2 4
3 9
Once we have the vertex, the rest of the points
follow the pattern Over 1, up 1 Over 2, up
4 Over 3, up 9
We can plot these points to get the PARABOLA
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Transformations of y x2
That parabola (which represents the simplest
quadratic function), can be
Reflected
Stretched vertically
Slid horizontally
Slid vertically
These are called transformations Ref, VS, HT,
VT,
The form y ax2 bx c is called the GENERAL
FORM
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Transformations
Ex. Graph the function
What are the transformations?
So we can use these to get a MAPPING RULE
Reflection No (the left side is not multiplied
by -1) Vertical Stretch, VS? Yes VS
2 Vertical Translational, VT? Yes VT
-3 Horizontal Translational, HT? Yes HT 1
(x,y) ? (x1, 2y-3)
We now take the old table of values and do what
the mapping rule says.
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Transformations
Ex. Graph the function
y x2 y x2
x y
-3 9
-2 4
-1 1
0 0
1 1
2 4
3 9
½(y3) (x-1)2 ½(y3) (x-1)2
x y
-2 15
-1 5
0 -1
1 -3
2 -1
3 5
4 15
So we can use these to get a MAPPING RULE
(x,y) ? (x1, 2y-3)
We now take the old table of values and do what
the mapping rule says.
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Transformations
Ex. Graph the function
½(y3) (x-1)2 ½(y3) (x-1)2
x y
-2 15
-1 5
0 -1
1 -3
2 -1
3 5
4 15
y x2 y x2
x y
-3 9
-2 4
-1 1
0 0
1 1
2 4
3 9
New Vertex (1,-3) that is (HT, VT)
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Transformations
½(y3) (x-1)2 ½(y3) (x-1)2
x y
-2 15
-1 5
0 -1
1 -3
2 -1
3 5
4 15
Ex. Graph the function
Once we find the new vertex, the rest of the
points follow the pattern Over 1, up 1 x VS Over
2, up 4 x VS Over 3, up 9 x VS
New Vertex (1,-3) that is (HT, VT)
Domain x e R Range y e -3,8) Axis of symmetry
x 1
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• Graph the following. State the axis of symmetry,
  range, domain and the y-intercept of each.
• A)
• B)
• C)

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A)
Domain x e R Range y e 3,8) Axis of symmetry
x 2 Y-int (0,11)
Ref No VS 2 VT 3 HT 2
(x,y)?(x2,2y3)
½(y-3) (x-2)2 ½(y-3) (x-2)2
x y
-1 21
0 11
1 5
2 3
3 5
4 11
5 21
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B)
Domain x e R Range y e (-8,0 Axis of symmetry
x -1 Y-int (0,-0.5)
Ref Yes VS 1/2 VT 0 HT -1
(x,y)?(x-1,-0.5y)
-2y (x1)2 -2y (x1)2
x y
-4 -4.5
-3 -2
-2 -0.5
-1 0
0 -0.5
1 -2
2 -4.5
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C)
Domain x e R Range y e (8,4 Axis of symmetry
x -6 Y int (0,-32)
Ref Yes VS 1 VT 4 HT -6
(x,y)?(x-6,-y4)
-(y-4) (x6)2 -(y-4) (x6)2
x y
-9 -5
-8 0
-7 3
-6 4
-5 3
-4 0
-3 -5
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Going backwards

• Find the equation of this graph

Ans 2(y1)(x-1)2
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Going Backwards

• What quadratic function has a range of y e(-8,6,
  a VS of 4 and an axis of symmetry of x -5?
• What quadratic function has a vertex of (2,-12)
  and passes through the point (1,-9)?

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Solving for x using transformational form
We can use transformational form to solve for x-
values given certain y-values.
Example The parabola
passes through the point (x,2). What are the
values of x?
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Solving for x using transformational form
Find the x-intercepts of the following quadratic
functions
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Changing Forms

• General Form y ax2 bx c
• Transformational Form
• Standard Form

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What are you good for?
Transformational Form
General Form f(x)y ax2 bx c
Good for
Finding the vertex (h,k)
Good for
Finding the y-intercept (0,c)
Getting the range
Finding the x-intercept(s) (soon)
Getting the axis of symmetry
Getting the max or min value (y-value of
the vertex)
Finding the vertex (soon)
Getting the mapping rule
Standard Form
Graphing the functions parabola
Getting the equation of the graph
Good for
Nothing
Draw backs Functions are usually written in the
f(x) notation.
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Going To General Form
Whenever you go TO GENERAL Form FOIL and SOLVE
FOR Y
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Going To General Form
FOIL
irst
utside
nside
ast
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Going to General Form