Question

The vertex form of the equation of a vertical parabola is given by , where (h, k) is the vertex of the parabola and the absolute value of p is the distance from the vertex to the focus, which is also the distance from the vertex to the directrix. You will use the GeoGebra geometry tool to create a vertical parabola and write the vertex form of its equation. Open GeoGebra, and complete each step below. If you need help, follow these instructions for using GeoGebra. Mark the focus of the parabola you are going to create at F(6, 4). Draw a horizontal line that is 6 units below the focus. This line will be the directrix of your parabola. What is the equation of the line?

Part F
What is the value of p for your parabola?


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Part G
Based on your responses to parts C and E above, write the equation of the parabola in vertex form. Show your work.


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Part H
Construct the parabola using the parabola tool in GeoGebra. Take a screenshot of your work, save it, and insert the image below.


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Part I
Once you have constructed the parabola, use GeoGebra to display its equation. In the space below, rearrange the equation of the parabola shown in GeoGebra, and check whether it matches the equation in the vertex form that you wrote in part G. Show your work.


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Part J
To practice writing the equations of vertical parabolas, write the equations of these parabolas in vertex form:

focus at (-5, -3), and directrix y = -6
focus at (10, -4), and directrix y = 6.

Answer 1

Answer:

Step-by-step explanation:

Focus: (6,4)

Directrix lies 6 units below the focus, so the parabola opens upwards and focal length p = 6/2 = 3.

The equation of the directrix is y = -2.

The vertex is halfway between focus and directrix, at (6,1).

Equation of the parabola:

y = (1/(4p))(x-6)²+1 = (1/12)(x-6)²+1

Answer 2

The equation of the parabola is $y = \frac{1}{12}(x - 6)^2 + 1$

What are parabolas?

Parabolas are used to represent a quadratic equation in the vertex form

The given parameters are:

Focus = (6,4)

Directrix (x) = 6 units below the focus,

Start by calculating the focal length (p)

$p = \frac x2$

This gives

$p = \frac 62$

$p = 3$

Next, calculate the vertex as follows:

$(h,k) = (6,2/2)$

Simplify

$(h,k) = (6,1)$

The equation of the parabola is then calculated a:

$y = \frac{1}{4p}(x - h)^2 + k$

So, we have:

$y = \frac{1}{4*3}(x - 6)^2 + 1$

Simplify

$y = \frac{1}{12}(x - 6)^2 + 1$

Hence, the equation of the parabola is $y = \frac{1}{12}(x - 6)^2 + 1$

Read more about parabola at:

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