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.
