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Parabola - Graphing

• Recall that the equations for a parabola are given by ...

)ky(p4)hx( 2 Vertical Axis of Symmetry

0p

0p

Horizontal Axis of Symmetry

0p

0p

)hx(p4)ky( 2

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Parabola - Graphing

• The vertex of the parabola is located at the point (h, k).

• Example 1: Determine the vertex of the parabola given by ...

)2x(4)3y( 2

The vertex is at V(2, -3).

• Geometrically, the vertex is the midpoint of the line segment joining the focus and the directrix.

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Parabola - Graphing

• Note that when the squared term is in x, the axis of symmetry is vertical, and the parabola is facing up or down.

)ky(p4)hx( 2 Vertical Axis of Symmetry Horizontal Axis of Symmetry

)hx(p4)ky( 2

• When the squared term is in y, the axis of symmetry is horizontal, and the parabola is facing left or right.

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Parabola - Graphing

• The value of p is the directed distance from the vertex to the focus.

• Example 2: Consider the parabola at the right.

• Since the parabola is facing down, the axis of symmetry is vertical and the equation is of the form ...

)ky(p4)hx( 2 • The distance from the vertex to the focus moves in a negative direction, implying that p 0.

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Parabola - Graphing

• Since the vertex is equidistant from the focus and the directrix (as are all points on the parabola by the definition), the distance from the vertex to the directix is | p | units.

• To determine the basic shape of the parabola, it is a good idea to plot one or more points other than the vertex.

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V (1, - 2)

Slide 6

Parabola - Graphing

• Example 3: Sketch the graph of the parabola whose equation is given at the right.

)1x(12)2y( 2

• The vertex is at V(1, -2).

4

4

- 4

• Since the square is on the y, the axis of symmetry is horizontal, with the parabola facing left or right ...

axis of symmetry y = - 2

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Parabola - Graphing

4

4

- 4

• Find the value of p ... )1x(12)2y( 2

3p

12p4

• Since p is negative, the parabola is facing left.

• Plot another point. Letting y = 4 yields x = -2, or the point (-2, 4) ...

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Parabola - Graphing

4

4

- 4

• Sketch the graph using the points and the axis of symmetry ...

• Using p = -3 the focus is three units to the left of the vertex ...

F (- 2, - 2)

• The directrix is a vertical line 3 units on the other side of the vertex.

directrix x = 4

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Parabola - Graphing