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Quadratic Functions and Transformations. Standard Form of a
Quadratic Function.
To identify and graph quadratic functions. To describe and graph
transformations of functions. To identify the axis of symmetry,
vertex, domain, and range for a given parabola in the vertex form
and the standard form of the quadratic function.
Parabola, quadratic function, vertex form, axis of symmetry,
vertex of the parabola, minimum value maximum value
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Take G.C.
53x2y
x2
1y
x7y
5xy
2xy
5xy
3xy
xy
2
2
2
2
2
2
2
2
Identify key features a. Intercepts b. b. intervals where the
function is increasing or decreasing, c. intervals where the
function is positive or negative; d. relative maximums and
minimums; e. symmetries; f. end behavior
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Take a note:
A parabola is the graph of a quadratic function, which you can
write in the form Remember this is a transformation of the parent
function
cbxaxxf 2)(
2)( xxf
The vertex form of the quadratic function is
khxaxf 2)()(
Reflection, stretch and compression
Translation in the x-axis
Translation on the y-axis
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Your turn 2x4
1y
2.Graph each function. How is each graph a translation of
2)3x()x(g)a
1x)x(h)b 2
Do a table of values or get the vertex: (0,0) and 2 units to the
right and left of the vertex. a‹0 reflection on x-axis 0‹a‹1
compression
Vertex (3,0), 3 units right
Vertex:(0,1), 1 unit up
b)
1. What is the graph of ?
a)
2x)x(f
Answer
Answer
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3. For .What are the vertex, the axis of symmetry, the minimum
or maximum value, the domain, and range?
5)3(2
1 2 xy
4.The arch of the Sidney Harbor Bridge is approximately 500
meters long and 85 meters high. What quadratic function models the
curve of the arch? Assume the arch starts at (0, 0).
Vertex (3,-5) Axis of symmetry x=3 a=1/2 compression Minimum on
y=-5
)85,250(vertex
)0,500(and)85,250(and)0,0(
k)hx(ay 2
85)250x(12500
17y 2
5y,Ry,yRange
Rx,xDomain
Answer
Answer 85)2500(a02
85a625000
a12500
17
62500
85
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cbxaxxf 2)(
Note
khxaxf 2)()(
Standard form Vertex Form
vertex vertex
)2
(,2
(a
bf
a
b (h,k)
Axis of symmetry Axis of symmetry
a
bx
2 x=h
Y-intercept is (0,c)
a has the same meaning for both
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Converting standard form to vertex form.
What is the vertex form of ? 7x10x2y 2
a=2 b=10 c=7
a2
bx
5.24
10
)2(2
10x 5.5
7)5.2(10)5.2(2
7102
2
2
y
y
xxy
The vertex is (-2.5,-5.5) 5.5)5.2x(2y 2
Your turn
What is the vertex form of 5x4xy 2 Answer:
1)2( 2 xy
Answer
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Interpreting a quadratic graph The new River George Bridge in
West Virginia is the world’s largest steel single arch bridge. You
can model the arc with the function where x and y are in feet. How
high above the river is the arch? How long is the section of the
bridge above the arch?
x847.0x000498.0y 2
a=-0.000498 b=0.847
1. Find the vertex
850)000498.0(2
847.0
a2
bx
360)850(847.0)850(000498.0y 2
Answer
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2.Find the height of the arch above its supports
516+360=876 ft
3. Find the height of the arch above the river
About 360 ft
4. Find the length of the bridge above the arch
850+850=1700 ft long
vertex )360,850(
Answer
Answer
Answer
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Your turn The Zhaozhou Bridge in China is the oldest known arch
bridge, dating to A.D.605. You can model the support arch with the
function .Where x and y are measured in feet. How high is the arch
above the supports
x131148.0x001075.0)x(f 2
61)001075.0(2
131148.0
a2
bx
a= - 0.001075 b= 0.131148
4999953.3)61(f
)61(131148.0)61(001075.0)61(f 2
Answer: 4 ft
Answer
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Classwork odd Homework even Text book pages 199-200 exercises
7-56 Text book pages 206-207 exercises 8-47
