I am creating a project for math in which I have to draw an object using: parabolas, hyperbolas, circles,etc.
We use conic section equations to identify where they are on the graph.
I have confusion about writing the equation of parabolas. I know the equation is 4px=y^2 but I can't find p because I don't know were the focus point is. I drew parabolas, how would I find the focus point (or directrix) to help me find p for the equation?
note: they don't give you an equation you need to make one yourself.
my vertex (11,3)
if the coefficient of x^2 is positive, the parabola will go up. Otherwise, it will go downwards
To find the focus point (or directrix) of a parabola given its vertex and equation, you can follow these steps:
Step 1: Determine the type of parabola
To find the focus point and directrix of a parabola, you'll first need to determine its orientation. Are you working with a vertical or horizontal parabola? This information can help you visualize and understand the shape of the parabola better.
In your case, since you have the equation 4px = y^2, we can deduce that you are working with a horizontal parabola.
Step 2: Identify the vertex
You've mentioned that the vertex of your parabola is at (11, 3). The vertex is always in the form (h, k), where h corresponds to the x-coordinate and k corresponds to the y-coordinate. So in this case, h = 11 and k = 3.
Step 3: Determine the distance
The distance between the vertex and the focus point (or directrix) is denoted as p in the equation 4px = y^2. To find p, we need to consider the values of h and k.
For a horizontal parabola:
- The focus point is located at (h + p, k), and the directrix is given by the equation x = h - p.
Step 4: Substitute the given vertex coordinates
Now you can substitute the vertex coordinates into the formulas to find the focus point and directrix equation for your parabola.
For the focus point:
Since the vertex is at (11, 3), the focus point is at (h + p, k), which becomes (11 + p, 3). While you don't have a value for p yet, you can keep it as a variable for now.
For the directrix:
The directrix equation for a horizontal parabola is x = h - p. Substituting the values h = 11 and k = 3, the equation becomes x = 11 - p.
Step 5: Analyze the parabola and determine p
Now that you have the equation for the directrix and the coordinates for the focus point, you can solve for p using the information you have about the parabola.
Remember, p represents the distance from the vertex to the focus point. In your case, the focus point, when written as an ordered pair, is (11 + p, 3). Additionally, the equation for the directrix is x = 11 - p.
Take some time to analyze the given information, such as the shape of the parabola, the vertex, and any additional constraints or details you might have. By combining this information with the equation for a parabola and the formulas for finding p, you can determine the appropriate value for p to complete the equation of the parabola.
Note: If you have any additional information or constraints for your parabola, please provide them, and I can assist you further in finding the value of p and completing the equation.