Find the standard form of the equation of the parabola with the given characteristics.
Vertex: (-9, 8); directrix: x = -16
plug and chug
(y-k)^2 = 4a (x-h)
k = 8 given
h = -9 given
distance from vertex to directrix = a
distance from -9 to - 16 = 7 = a (positive for directrix to the left of the vertex)
so
(y-8)^2 = 28 (x+9)
Focus: (1515,00); Directrix: xequals=negative 15
Focus: (15,0); Directrix: x= -15
To find the standard form of the equation of a parabola given its vertex and directrix, you can follow these steps:
Step 1: Understand the characteristics of the parabola:
- The vertex of the parabola is given as (-9, 8).
- The directrix is x = -16.
Step 2: Understand the standard form of the equation of a parabola:
The general standard form of the equation of a parabola is: (x - h)^2 = 4p(y - k)
In this equation, (h, k) is the vertex of the parabola, and p is the distance between the vertex and the focus (or directrix).
Step 3: Find the value of p:
To find p, we need to calculate the distance between the vertex and the directrix. In this case, the directrix is a vertical line x = -16, and the vertex is at (-9, 8).
Since the directrix is vertical, the distance between the vertex and the directrix is the absolute value of the difference between the x-coordinates of the vertex and the directrix:
p = |-9 - (-16)| = |-9 + 16| = |7| = 7
Step 4: Write the equation in standard form:
Now that we have the values of h, k, and p, we can substitute them into the standard form equation: (x - (-9))^2 = 4 * 7(y - 8)
Simplifying the equation, we get: (x + 9)^2 = 28(y - 8)
So, the standard form of the equation of the parabola with the given characteristics is (x + 9)^2 = 28(y - 8).