write the equation for and graph the parabola.
Focus:(-6,7)
Opens:Left
Contains:(-6,5)
X-AXIS=-6
Y-AXIS=7 AND
SIMILAR THE LAST TWO NO.S
To determine the equation of a parabola, we need to consider the focus, the direction it opens, and a point on the parabola.
Since the given parabola opens to the left, its general equation can be written as:
(x - h)^2 = -4p(y - k)
where (h, k) represents the vertex. In this case, the vertex can be determined using the focus and the direction of the opening.
Given that the focus is (-6, 7), and the parabola opens to the left, we can determine the vertex as (k, h) = (7, -6).
Now, we just need to find the value of p. To do this, we can use the distance formula between the focus and the vertex, knowing that p represents the distance between them.
The formula for the distance between two points is:
d = sqrt((x2 - x1)^2 + (y2 - y1)^2)
Substituting the known values, we have:
p = sqrt((-6 - -6)^2 + (7 - 5)^2) = sqrt(0 + 4) = 2
Now we have all the information needed to write the equation:
(x + 6)^2 = -4(2)(y - 7)
Simplifying, we find:
(x + 6)^2 = -8(y - 7)
To graph the parabola, we can plot the vertex and use the value of p to find additional points. Since the parabola opens to the left, we can use the value of p to find points to the right of the vertex.
Plotting the vertex (-6, 7), we can find additional points by subtracting p from the x-coordinate:
(-6 - 2, 7) = (-8, 7)
We can continue this process to find more points. For example, subtracting 2 from x again:
(-6 - 2 - 2, 7) = (-10, 7)
Plotting these points and connecting them, we get the graph of the parabola.