Answered

IDNLearn.com is the perfect place to get detailed and accurate answers to your questions. Our platform offers comprehensive and accurate responses to help you make informed decisions on any topic.

Question 1

The vertex form of the equation of a vertical parabola is given by
[tex]\[ y = \frac{1}{4p}(x-h)^2 + k, \][/tex]
where [tex]\((h, k)\)[/tex] is the vertex of the parabola and the absolute value of [tex]\(p\)[/tex] is the distance from the vertex to the focus, which is also the distance from the vertex to the directrix.

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 the instructions for using GeoGebra.

Sagot :

GinnyAnsweridn
To create a vertical parabola and write its vertex form equation using GeoGebra, follow these steps:

1. Open GeoGebra Tool:
Launch GeoGebra and select the geometry tool from the interface.

2. Plot the Vertex:
Using the "Point" tool, plot the vertex of the parabola, say at point [tex]\( (h, k) \)[/tex].

3. Define the Focus:
Determine the location of the focus. For instance, if the vertex is at [tex]\( (h, k) \)[/tex] and the focus is [tex]\( p \)[/tex] units above the vertex, plot the focus at [tex]\( (h, k + p) \)[/tex].

4. Draw the Parabola:
Select the "Parabola" tool. Click on the vertex point [tex]\( (h, k) \)[/tex] first, and then click on the focus point [tex]\( (h, k + p) \)[/tex]. This will sketch the parabola for you.

5. Find the Value of [tex]\( p \)[/tex]:
Measure the distance [tex]\( p \)[/tex] from the vertex to the focus. Use the "Distance or Length" tool to do so. This gives you the value of [tex]\( p \)[/tex].

6. Write the Vertex Form Equation:
Use the measured values [tex]\( h \)[/tex], [tex]\( k \)[/tex], and [tex]\( p \)[/tex] to write the equation of the parabola in its vertex form [tex]\( y = \frac{1}{4p}(x - h)^2 + k \)[/tex].

7. Example Detailed Solution:
Let's assume you have plotted the vertex at [tex]\( (2, 3) \)[/tex] and placed the focus 2 units above the vertex, at point [tex]\( (2, 5) \)[/tex]:

- The vertex of the parabola [tex]\( (h, k) \)[/tex] is [tex]\( (2, 3) \)[/tex].
- The distance [tex]\( p \)[/tex] from the vertex to the focus is 2 units.

Substitute these values into the vertex form equation:

[tex]\[ y = \frac{1}{4p}(x - h)^2 + k \][/tex]

- Here, [tex]\( h = 2 \)[/tex], [tex]\( k = 3 \)[/tex], and [tex]\( p = 2 \)[/tex].
- Therefore, the equation becomes:

[tex]\[ y = \frac{1}{4 \cdot 2}(x - 2)^2 + 3 \][/tex]

Simplify:

[tex]\[ y = \frac{1}{8}(x - 2)^2 + 3 \][/tex]

So, the final vertex form of the equation of the parabola is:

[tex]\[ y = \frac{1}{8}(x - 2)^2 + 3 \][/tex]

By following these steps, you can use GeoGebra to create the graphical representation of a vertical parabola and determine its vertex form equation.
Thank you for using this platform to share and learn. Don't hesitate to keep asking and answering. We value every contribution you make. Your questions deserve reliable answers. Thanks for visiting IDNLearn.com, and see you again soon for more helpful information.