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Consider three points: and

Determine the distance between each point and .

Determine the distance between each point and the line,

𝑪(−1.5 ,2.25)

𝑩(2 ,4 )

𝐴(1 ,1)

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Precalculus

The PARABOLAConic Sections:

Von Christopher G. Chua

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PARABOLAS:Equations and Graphs

In the next two sessions, you are expected to develop the ability to…

OUR LEARNING

GOALS

1. define a parabola;2. Determine the standard

form of equation of a parabola;

3. Graph a parabola in a rectangular coordinate system; and

4. Solve situational problems involving parabolas.

This slideshow presentation will be made available through the course website: mathbychua.weebly.com.Download the document to use it as reference.

DEFINITION PARABOLA

Let be a given point, and ℓ a given line not containing . The set of all points such that its distances from and from ℓ are equal, is called a parabola. The point is its focus and the line ℓ its

directrix.

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EQUATION of a PARABOLA

𝑃 (𝑥 , 𝑦)

𝐷(𝑥 ,−𝑐)

𝐹 (0 ,𝑐)

The standard form of the equation of a parabola

with vertex is at the point of origin and opens

upward or downward is

If a parabola with its vertex at the opens

sideways, the standard form of its equation is verte

x

axis of symmetry

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𝑥2=4𝑐𝑦 ,𝑐>0

When the parabola opens

upward

𝑥2=4𝑐𝑦 ,𝑐<0

When the parabola opens

downward

𝑦 2=4𝑐𝑥 ,𝑐>0

When the parabola opens

to the right

𝑦 2=4𝑐𝑥 ,𝑐<0

When the parabola opens

to the left

TYPES of PARABOLA4

1 2

3 4

SIDE QUESTION:

What do you notice about the position of the

focus with respect to the

graph?

LET’S DISCUSS

To which direction does each of the following parabolas open to?

Determine the Orientation

EXAMPLE

How do we determine the focus and directrix of a parabola with vertex at the origin?

Since the quadratic variable is and the coefficient of is positive, the parabola opens upward.

The vertex is at and the axis of symmetry is the -axis or

Compared to , we can determine that The focus of the parabola is therefore at and its directrix

is the line

Focus & Directrix

YOUR TURN!

Determine the focus and directrix of the following parabola based from the given equation.

Focus & Directrix

WHAT IF…

But what if the parabola does not have its vertex at the

point of origin?

!

PARABOLAS with Vertex NOT on

WHAT IF…

The standard form of the equation of a parabola that opens upward or

downward is

If a parabola opens sideways, the standard form of its equation is

(𝑥−h)2=4𝑐 (𝑦−𝑘)

(𝑦−𝑘)2=4 𝑐(𝑥−h)

v

EXAMPLE

Describe the parabola, .What is its graph’s orientation?What are the coordinates of its vertex?What is the value of in the equation?So, if , what are the coordinates of the focus?What is the equation of the line that is the directrix?

The graph opens upward.Its vertex is at If , then The coordinates of the focus are The directrix is .

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v

focus

directrix

Axis of symmetry

v

focus

directrix

Axis of symmetry