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Chapter 6 Section 1Chapter 6 Section 1

Graphing Quadratic Functions

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Chapter 6 Section 1Chapter 6 Section 1

1.1. Find Find a = , b = , ca = , b = , c = . = .

2.2. Find Find y intercepty intercept = (0, = (0, c c).).

3.3. Find Axis of SymmetryFind Axis of Symmetry

4.4. Find Vertex ( AOS , __ ) Find Vertex ( AOS , __ )

Plug Plug AOSAOS in function to find in function to find y.y.

5.5. Look at Look at a a is it (+)min or (-)maxis it (+)min or (-)max

6.6. Find Value Max/Min (Find Value Max/Min (yy of vertex). of vertex).

7.7. Make Table of Values and Plot put Make Table of Values and Plot put vertexvertex in the in the center of the table and graph.center of the table and graph.

a

bx

2

A A quadratic function quadratic function is described by an is described by an equation of the following form.equation of the following form.

cbxaxxf 2

Linear termLinear term

Quadratic termQuadratic term Constant termConstant term

The graph of any quadratic function is called a The graph of any quadratic function is called a parabolaparabola....

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Chapter 6 Section 1Chapter 6 Section 1

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Chapter 6 Section 1Chapter 6 Section 1

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Chapter 6 Section 1Chapter 6 Section 1

Find the Find the yy-intercept, the equation of the axis of -intercept, the equation of the axis of symmetry, the vertex , Max or min and Value, then symmetry, the vertex , Max or min and Value, then graph. graph. ff((xx) = ) = xx22 + 9 + 8 + 9 + 8xx

Step 1: Arrange terms. Then identify Step 1: Arrange terms. Then identify a, b, and c a, b, and c ff((xx) = ) = xx22 + 9 + 8 + 9 + 8xxff((xx) = ) = xx22 + 8 + 8x +x + 9 9

So So aa = 1, = 1, bb = 8, and = 8, and cc = 9 = 9

Step 2: Find the Step 2: Find the yy-intercept, (0, c) -intercept, (0, c) The The yy-intercept is (0, 9).-intercept is (0, 9).

Step 3: Find the Axis of Symmetry (AOS)Step 3: Find the Axis of Symmetry (AOS)

x b

2a

x 8

2 1 4 AOS = -4AOS = -4

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Chapter 6 Section 1Chapter 6 Section 1

Find the Find the yy-intercept, the equation of the axis of -intercept, the equation of the axis of symmetry, the vertex , Max or min and Value, then symmetry, the vertex , Max or min and Value, then graph. graph. ff((xx) = ) = xx22 + 8 + 8x x + 9+ 9

Step 4: Find the coordinates of the vertex. (AOS, ___). Step 4: Find the coordinates of the vertex. (AOS, ___). Plug AOS in original function to find Plug AOS in original function to find y - y - coordinatecoordinateff(-4) = (-4) = xx22 + 8 + 8x +x + 9 9 = (-4)= (-4)22 + 8(-4) + 9 = 16 - 32 + 9 = -7 + 8(-4) + 9 = 16 - 32 + 9 = -7

Step 5: Max or Min Step 5: Max or Min aa = 1, positive so Minimum = 1, positive so MinimumStep 6: Value of Max/Min: Step 6: Value of Max/Min:

(-4, -7)(-4, -7)vertexvertex

––7 7

Min: –7 Min: –7

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Chapter 6 Section 1Chapter 6 Section 1

xx xx22 + 8 + 8x + x + 99 ff((xx)) ((x,x, ff((xx))

-6-6

-5-5

-4-4

-3-3

-2-2

Find the Find the yy-intercept, the equation of the axis of -intercept, the equation of the axis of symmetry, the vertex , Max or min and Value, then symmetry, the vertex , Max or min and Value, then graph. graph. ff((xx) = ) = xx22 + 8 + 8x + x + 99

xx xx22 + 8 + 8x + x + 99 ff((xx)) ((x,x, ff((xx))

-6-6 (-6)(-6)22 + 8(-6) + 9 + 8(-6) + 9 -3-3 (-6, -3)(-6, -3)

-5-5 (-5)(-5)22 + 8(-5) + 9 + 8(-5) + 9 -6-6 (-5, -6)(-5, -6)

-4-4 (-4)(-4)22 + 8(-4) + 9 + 8(-4) + 9 -7-7 (-4, -7)(-4, -7)

-3-3 (-3)(-3)22 + 8(-3) + 9 + 8(-3) + 9 -6-6 (-3, -6)(-3, -6)

-2-2 (-2)(-2)22 + 8(-2) + 9 + 8(-2) + 9 -3-3 (-2, -3)(-2, -3)

vertexvertex

(-4, -7)(-4, -7)vertexvertex

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Chapter 6 Section 1Chapter 6 Section 1

Graph Graph ff((xx) = ) = xx22 + 8 + 8xx + 9 + 9

((x,x, ff((xx))

(-6, -3)(-6, -3)

(-5, -6)(-5, -6)

(-4, -7)(-4, -7)

(-3, -6)(-3, -6)

(-2, -3)(-2, -3)

AOS AOS x x = -4= -4

x x = -4= -4yy-intercept -intercept

(0, 9)(0, 9)

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Chapter 6 Section 1Chapter 6 Section 1

Consider the function f(x) = x2 – 4x + 9

To find Max/Min without graphing do Steps 1 – 6.To find Max/Min without graphing do Steps 1 – 6.

Step 1. Step 1. aa = 1, = 1, bb = = – – 4, and 4, and cc = 9 = 9

Step 2. Step 2. yy–intercept (0, 9)–intercept (0, 9)

Step 3.Step 3.

Step 4. Find Vertex (2, __)Step 4. Find Vertex (2, __)

ff(2) = (2)(2) = (2)22 - 4(2) + 9 = 4 - 8 + 9 = 5 - 4(2) + 9 = 4 - 8 + 9 = 5

Step 5. Max/Min? Step 5. Max/Min? aa = 1. = 1. a a is positive minimum value.is positive minimum value.

Step 6. Value of Max/Min. The Vertex is (2, 5) So theStep 6. Value of Max/Min. The Vertex is (2, 5) So the

Min value is 5Min value is 5..

AOS b2a

4 2(1)

2 2

55

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Chapter 6 Section 1Chapter 6 Section 1

1.1. Find Find a = , b = , ca = , b = , c = . = .

2.2. Find Find y intercepty intercept = (0, = (0, c c).).

3.3. Find Axis of SymmetryFind Axis of Symmetry

4.4. Find Vertex ( AOS , __ ) Find Vertex ( AOS , __ )

Plug Plug AOSAOS in function to find in function to find y.y.

5.5. Look at Look at a a is it (+)min or (-)maxis it (+)min or (-)max

6.6. Find Value Max/Min (Find Value Max/Min (yy of vertex). of vertex).

7.7. Make Table of Values and Plot put Make Table of Values and Plot put vertexvertex in the in the center of the table and graph.center of the table and graph.

a

bx

2

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Chapter 6 Section 1Chapter 6 Section 1

Chapter 6 Section 1Skill Practice Workbook (SPW) Pg 36 All