Find the vertex, Focus,and Directrix of the parabola. Graph the equation.
y^2=12x
compare y^2 = 12x with y^2= 4px
p = 3
vertex (0,0)
Focus (3,0)
directric x = -3
To find the vertex, focus, and directrix of the parabola y^2 = 12x, we can use the standard form for a parabola with the vertex at the origin.
Step 1: Vertex
The vertex of the parabola y^2 = 12x is at the origin (0, 0).
Step 2: Focus
The focus of a parabola with equation y^2 = 4ax is located at the point (a, 0), where "a" is the coefficient of x. In this case, "a" = 3, so the focus is at (3, 0).
Step 3: Directrix
The directrix of a parabola with equation y^2 = 4ax is the vertical line that is a distance of "a" from the vertex and on the opposite side of the vertex. In this case, the directrix is the line x = -3.
Step 4: Graphing
To graph the parabola, plot the vertex at the origin (0, 0). Also, plot the focus at (3, 0). Draw a vertical line at x = -3, indicating the directrix.
The graph of the equation y^2 = 12x is a right-facing parabola:
To find the vertex, focus, and directrix of a parabola, we can start by rewriting the equation in the standard form:
y^2 = 4px
In this case, the equation is y^2 = 12x. Comparing it to the standard form equation, we can see that 4p = 12, which means p = 3.
The vertex of the parabola is given by the coordinates (h, k), where h = 0 and k = 0. So the vertex is at (0, 0).
To find the focus of the parabola, we use the formula:
F = (h + p, k)
In this case, substituting the values, we get:
F = (0 + 3, 0) = (3, 0)
Therefore, the focus of the parabola is at (3, 0).
To find the directrix of the parabola, we use the equation:
x = h - p
Substituting the values, we get:
x = 0 - 3 = -3
Therefore, the directrix of the parabola is at x = -3.
To graph the equation y^2 = 12x, we can start by plotting the vertex, focus, and directrix.
- Plot the vertex at (0, 0).
- Plot the focus at (3, 0).
- Draw a vertical line passing through the focus.
- The directrix is the line x = -3, parallel to the x-axis.
Now, we can draw the parabola symmetrically on both sides of the y-axis, curving towards the focus and touching the directrix at all points.
I hope this helps! Let me know if you have any further questions.