Table of Contents

Parabola - Finding the Equation

• 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 - Finding the Equation

• Summary of meanings of the constants:

The vertex is at V(h, k). and is the midpoint of theline segment joining the focus and the vertex.

When the squared term is in x, the axis ofsymmetry is vertical, and the parabola is

facing up or down. If p is positive, then up.If p is negative, then down.

When the squared term is in y, the axis of symmetry is horizontal, and the parabola is facing left or right. If p is positive, then

right. If p is negative, then left.

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Parabola - Finding the Equation

• Example 1: Find the equation of the parabola with focus F(-3, -2) and directrix y = 6.

-10

-8

-6

-4

-2

0

2

4

6

8

10

-8 -6 -4 -2 2)2,3(F

)2,3(V • Plot the focus point and draw the line for the directrix ...

6y

• The focus is 8 units from the directrix, half of which is 4 units. Thus, the vertex is at V(-3, 2).

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Parabola - Finding the Equation

-10

-8

-6

-4

-2

0

2

4

6

8

10

-8 -6 -4 -2 2)2,3(F

)2,3(V

6y

• Since a parabola faces towards the focus and away from the directrix, this parabola is facing down, and the equation is ...

)ky(p4)hx( 2 • The value of p is - 4 ...

p

• Substituting values yields ...

2( 3) 16( 2)x y

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Parabola - Finding the Equation

• Example 2: Find the standard form of the parabola given by ...

04y8y2x4 2 • Solution:

04y8y2x4 2

4x4y8y2 2

4x4)y4y(2 2

84x4)4y4y(2 2

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Parabola - Finding the Equation

84x4)4y4y(2 2

4x4)2y(2 2

)1x(4)2y(2 2

)1x(2)2y( 2

• The standard form of the parabola is ...

Table of Contents

Parabola - Finding the Equation