Graph the parabolas and identify the axis, directrix, and focus: (y^2)+8x=0
I know that the axis refers to the axis of symetry and the directrix is the same distance away from the vertex as the foucus is. I just don't know how to graph and identify all the parts from an equation like the one above.
Recall that the parabola
y^2 = 4px
has
vertex at (0,0)
focus at (p,0)
directrix at x = -p
(y^2)+8x=0
can be written as
y^2 = -8x
so p = -2
now just plug and chug.
It is clearly a parabola opening to the left with vertex at (0,0).
Thank you so much Steve!! just two questions, what does the variable "p" stand for/ represent and could I be able to represent the axis of symetry as an equation/point.
wait dose the axis even mean the axis of symetry??
yes, "axis" means the axis of symmetry.
and yes, your parabola has the x-axis as its axis of symmetry.
y = 0
thanks you so much!!
Can someone help me please.
How to find the equation of the axis of parabola and its length of the latus rectum having y-x²+8x=0?
To graph the parabola and identify the axis, directrix, and focus, follow these steps:
Step 1: Rewrite the equation in the standard form of a parabola, which is (x - h)^2 = 4p(y - k), where (h, k) is the vertex coordinates and p is the distance between the vertex and the focus/directrix.
For the given equation (y^2)+8x=0, we need to complete the square to transform it into the standard form. Rearranging the equation, we have:
8x = -y^2
Dividing through by 8, we get:
x = (-1/8)y^2
Step 2: Compare the equation to the standard form. Notice that there is a negative coefficient for y^2, which means the parabola opens to the left.
Step 3: Identify the vertex. The vertex, denoted as (h, k), is at the point where the parabola is symmetric. In this case, since the parabola opens to the left, the vertex will have the form (k, h).
In the equation x = (-1/8)y^2, there is no x-term, which means the vertex is at (0, 0).
Step 4: Determine the distance between the vertex and the focus/directrix. The distance, denoted as p, is an important value in defining the parabola.
Since the parabola opens to the left, p is negative. To find the value of p, we can use the formula: p = -1/(4a), where a is the coefficient of the y^2 term.
In this case, a = -1/8, so substitute it into the formula:
p = -1 / (4 * (-1/8))
p = -1 / (-1/2)
p = 2
Step 5: Find the focus. The focus is a point on the axis of symmetry, located at a distance of p units from the vertex.
Since the vertex is at (0, 0), the focus will be at (-p, 0), which, in this case, is (-2, 0).
Step 6: Determine the directrix. The directrix is a line that is perpendicular to the axis of symmetry and is located at a distance of p units in the opposite direction from the vertex.
Since the vertex is at (0, 0), the directrix will be the vertical line x = p, which, in this case, is x = 2.
Step 7: Graph the parabola. With the vertex, focus, and directrix identified, you can now plot these points on the coordinate plane and sketch the parabola. Since the equation opens to the left, the parabola will be shaped like a bowl facing leftward.
Place the vertex at (0, 0), the focus at (-2, 0), and draw the directrix x = 2. Now, plot a few additional points on the parabola, such as (-4, 2), (-8, 4), (-12, 6), and connect them to form the curve.
The final graph should show the parabola with the identified axis, directrix, and focus as described above.