4-1 quadratic functions unit objectives: solve a quadratic equation. graph/transform quadratic...
TRANSCRIPT
4-1 Quadratic Functions
Unit Objectives:• Solve a quadratic equation.• Graph/Transform quadratic functions with/without a calculator• Identify function attributes from graph or equation• Model situations with quadratic functions.
Today’s Objective:• Identify attributes and graph quadratic functions
-5 5-5
5
x
y
x
y
x
y
Parent function/equation:
Quadratic Function:Graph:• ParabolaVertex
• Point where graph changes direction• Minimum or maximum• Vertex Form: • Vertex: (h, k)
Axis of Symmetry (line) • Divides the graph into 2 mirror images• x = h
where
𝑦=𝑥2
Transformation of Translation: Vertical Translation: Horizontal
Stretch or Compression: Reflection
𝑦=𝑥2+𝑘Up k units
Down k units
𝑦=𝑥2−𝑘
𝑦=(𝑥−h)2Right h units
Left h units
𝑎>1Stretch:
Shrink:
𝑦=𝑎𝑥2
𝑦=−𝑥2Across x-axis
0<𝑎<1
𝑦=(𝑥+h)2
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Vertex Form:
Graphing a Quadratic Function in vertex form1. Plot the vertex2. Find and plot two points
to the right of vertex. 3. Plot the point across axis
of symmetry.4. Sketch the curve.
Vertex:
Axis of Symmetry:Domain:
Range:
All Real Numbers
𝑦=(𝑥−3)2+2 (3 ,2)
-2 8-2
8
x
y
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y
Units right of vertex
x
Units up from
vertex
1
2
14
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y
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y
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y
𝑦 ≥ 2
𝑥2
𝑥=3
Graphing a Quadratic Function in vertex form1. Plot the vertex2. Find and plot two points
to the right of vertex. 3. Plot the point across axis
of symmetry.4. Sketch the curve.
Vertex:
Axis of Symmetry:Domain:
Range:
All Real Numbers
𝑦=2 𝑥2 (0 , 0)Units
right of vertex
x
Units up from
vertex
1
2
28
𝑦 ≥ 0
2 𝑥2
-5 5-2
8
x
y
x
y
x
y
x
y
x
y
𝑥=0
Graphing a Quadratic Function in vertex form1. Plot the vertex2. Find and plot two points
to the right of vertex. 3. Plot the point across axis
of symmetry.4. Sketch the curve.
Vertex:
Axis of Symmetry:Domain:
Range:
All Real Numbers
𝑦=−12(𝑥+4 )
2
− 3 (− 4 ,− 3)Units
right of vertex
x
Units up from
vertex
1
2
−12
−2
𝑦 ≤ − 3
−12𝑥
2
-8 2-8
2
x
y
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y
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𝑥=− 4
-8 2
-8
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Writing a Quadratic function: vertex form
Identify the Vertex:
Finding stretch factor:• Choose another known point and
solve for a.
(− 2 ,−7 )𝑦=𝑎(𝑥+2)2−7
−5=𝑎(− 1+2)2− 72=𝑎
𝑦=2(𝑥+2)2− 7
-8 2
-8
2
x
y
-8 2
-8
2
x
y
(-2, -7)
(-1, -5)
Writing a Quadratic function: vertex form
Identify the Vertex:
Finding stretch factor:• Choose another known point and
solve for a.
(3 ,9)𝑦=𝑎(𝑥− 3)2+9
7=𝑎(5− 3)2+9−2=4𝑎
𝑦=−12(𝑥− 3)2+9
-2 8
-2
10
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(3, 9)
(5, 7)
−12=𝑎Pg. 199
#7-37 odds