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Graphs parabolas by Graphs parabolas by

calculating strategic pointscalculating strategic points

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Strategic points to calculate

• Establish orientation of parabola

• Axis of Symmetry

• Vertex

• Roots

• y- intercept

• If you do not have 5 points substitute a value for x and calculate the corresponding y

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Parabolas

y = ax2 + bx + c

When the coefficient of x2 is positive the graph is -shaped.

When the coefficient of x2 is negative the graph is -shaped.

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Graphs of the form y = ax2 + bx + c

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Parabolas

The axis of symmetry is a vertical lineThe equation of a the axis of symmetry is EC

The vertex is located on the axis of symmetry – it has a x-coordinate of Find the y-coordinate by plugging in for x

Parabolas have a vertical axis of symmetry …

…and a turning point called the vertex.

a

bx

2

a

b

2

a

b

2

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Sketching graphs of quadratic functions

y = 2x2 – 5x – 3 c = – 3

The parabola crosses the y-axis at the point (0, –3).

The quadratic function y = ax2 + bx + c will cross the y-axis at the point (0, c).

The quadratic function y = ax2 + bx + c will cross the y-axis at the point (0, c).

Sketch the graph of the function y = x2 – 2x – 3.

Axis of symmetry: Vertex

1

)1(2

)2(2

x

x

a

bx

)4,1(

41

3)1(2)1(1 2

yandx

yandx

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Sketching graphs of quadratic functions

When a quadratic function factors we can use its factored form to find where it crosses the x-axis. For example:

Sketch the graph of the function y = x2 – 2x – 3.

The function crosses the x-axis when y = 0.

x2 – 2x – 3 = 0

(x + 1)(x – 3) = 0

x + 1 = 0 or x – 3 = 0

x = 3

The function crosses the x-axis at the points (–1, 0) and (3, 0).

x = –1

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Sketching graphs of quadratic functions

0

y

x

(–1, 0) (3, 0)

(0, –3)

(1, –4)

We can now sketch the graph.

• Establish orientation of parabola: open up

• Axis of Symmetry x=1• Vertex (1,-4)• y- intercept y=-3• Roots x=-1 or x=3

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Sketching graphs of quadratic functions

When a quadratic function is written in the form y = a(x – p)(x – q), it is called

factored form and p and q are the roots of the quadratic function.

When a quadratic function is written in the form y = a(x – p)(x – q), it is called

factored form and p and q are the roots of the quadratic function.

In general:

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Graphs of the form y = a(x – p)(x – q)

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You try

• Find

a)Establish orientation of parabola

b)Axis of Symmetry

c)Vertex

d)Roots

e)y- intercept

Then graph the parabola

443 2 xxy