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3.2 Graphing Quadratic Functions in Vertex or Intercept Form. Definitions 3 Forms Graphing in vertex form Examples Changing between eqn. forms. Quadratic Function. A function of the form y=ax 2 +bx+c where a ≠0 making a u-shaped graph called a parabola. Example quadratic equation:. - PowerPoint PPT Presentation
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3.2 Graphing Quadratic 3.2 Graphing Quadratic FunctionsFunctions in Vertex or in Vertex or
Intercept FormIntercept Form
• DefinitionsDefinitions
• 3 Forms3 Forms
• Graphing in vertex formGraphing in vertex form
• ExamplesExamples
• Changing between eqn. formsChanging between eqn. forms
Quadratic FunctionQuadratic Function•A function of the form A function of the form
y=axy=ax22+bx+c where a+bx+c where a≠0 making a ≠0 making a u-shaped graph called a u-shaped graph called a parabolaparabola..
Example quadratic equation:
Vertex-Vertex-
• The lowest or highest pointThe lowest or highest point
of a parabola.of a parabola.
VertexVertex
Axis of symmetry-Axis of symmetry-
• The vertical line through the vertex of the The vertical line through the vertex of the parabola.parabola.
Axis ofSymmetry
Vertex Form EquationVertex Form Equationy=a(x-h)y=a(x-h)22+k+k
• If a is positive, parabola opens upIf a is positive, parabola opens up
If a is negative, parabola opens If a is negative, parabola opens down.down.
•The vertex is the point (h,k).The vertex is the point (h,k).
•The axis of symmetry is the The axis of symmetry is the vertical line x=h.vertical line x=h.
Vertex FormVertex FormEach function we just looked at can be written Each function we just looked at can be written
in the form (x – h)in the form (x – h)22 + k, where (h , k) is the + k, where (h , k) is the vertex of the parabola, and x = h is its axis of vertex of the parabola, and x = h is its axis of symmetry.symmetry.
(x – h)(x – h)22 + k – vertex form + k – vertex formEquationEquation VertexVertex Axis of Axis of
SymmetrySymmetry
y = xy = x22 or or y = (x – y = (x – 00))22 + + 00
((00 , , 00)) x = x = 00
y = xy = x22 + 2 or + 2 ory = (x – y = (x – 00))22 + + 22
((0 0 , , 22)) x = x = 00
y = (x – y = (x – 33))22 or or y = (x – y = (x – 33))22 + + 00
((33 , , 00)) x = x = 33
Example 1: GraphExample 1: Graph
•Analyze y = (x + 2)Analyze y = (x + 2)22 + 1. + 1.• Step 1 Step 1 Plot the vertex (-2 , 1)Plot the vertex (-2 , 1)
• Step 2 Step 2 Draw the axis of symmetry, x = -Draw the axis of symmetry, x = -2.2.
• Step 3Step 3 Find and plot two points on one Find and plot two points on one side side , such as (-1, 2) and (0 , 5)., such as (-1, 2) and (0 , 5).
• Step 4Step 4 Use symmetry to complete the Use symmetry to complete the graph, or find two points ongraph, or find two points on
the left side of the vertex.the left side of the vertex.
Your Turn!Your Turn!
•Analyze and Graph:Analyze and Graph:
y = (x + 4)y = (x + 4)22 - 3. - 3.
(-4,-3)
Example 2: GraphExample 2: Graphy=-.5(x+3)y=-.5(x+3)22+4+4• a is negative (a = -.5), so parabola opens down.a is negative (a = -.5), so parabola opens down.• Vertex is (h,k) or (-3,4)Vertex is (h,k) or (-3,4)• Axis of symmetry is the vertical line x = -3Axis of symmetry is the vertical line x = -3• Table of values Table of values x y x y
-1 2-1 2 -2 3.5 -2 3.5
-3 4-3 4 -4 3.5-4 3.5 -5 2-5 2
Vertex (-3,4)
(-4,3.5)
(-5,2)
(-2,3.5)
(-1,2)
x=-3
Now you try one!Now you try one!
y=2(x-1)y=2(x-1)22+3+3
•Open up or down?Open up or down?
•Vertex?Vertex?
•Axis of symmetry?Axis of symmetry?
•Table of values with 5 points?Table of values with 5 points?
Changing from vertex or Changing from vertex or intercepts form to standard intercepts form to standard
formform• The key is to follow ORDER OF OPERATIONSThe key is to follow ORDER OF OPERATIONS
• Ex: y=-(x+4)(x-9)Ex: y=-(x+4)(x-9) Ex: y=3(x-1)Ex: y=3(x-1)22+8+8
=-(x=-(x22-9x+4x-36)-9x+4x-36) =3(x-1)(x-1)+8 =3(x-1)(x-1)+8
=-(x=-(x22-5x-36)-5x-36) =3(x =3(x22-x--x-x+1)+8x+1)+8
y=-xy=-x22+5x+36+5x+36 =3(x =3(x22--2x+1)+82x+1)+8
=3x=3x22-6x+3+8-6x+3+8
y=3xy=3x22-6x+11-6x+11
Changing from vertex or Changing from vertex or intercepts form to standard intercepts form to standard
formform• Practice:Practice:
• 1: y = 3(x-4)(x+2)1: y = 3(x-4)(x+2)
• 2: y = -2(x-3)2: y = -2(x-3)22 - 5 - 5