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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 :

GinnyAnswer
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.
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