write the parabola in y squared=4ax or x squared =4ax form where x equals the distance between the foci and the centerpoint
y= -(x+3)squared +4
i have no idea how to do this
can you tell me how you got the solution
Sure! To write the equation of a parabola in the form "y^2 = 4ax" or "x^2 = 4ay", we need to know the distance between the focus and the vertex.
First, let's understand the general form of a parabola equation. The standard form of a parabola is given by:
1. For a vertical parabola: (x - h)^2 = 4a(y - k)
2. For a horizontal parabola: (y - k)^2 = 4a(x - h)
Where (h, k) represents the vertex, and 'a' is a constant that determines the shape and orientation of the parabola.
Now, let's analyze the given equation: y = -(x + 3)^2 + 4
Here, we observe that the vertex of the parabola is (-3, 4) since the term inside the brackets (x + 3) shifts the vertex from the origin.
To determine the value of 'a', we need to know the distance between the vertex and the focus. In this case, since the parabola is vertical, the focus will be a point below the vertex. The distance between the vertex and the focus is equal to 'a'.
To find the value of 'a', we need to consider the standard form of the parabola equation for vertical parabolas:
(x - h)^2 = 4a(y - k)
Since the vertex is (-3, 4), the equation becomes:
(x + 3)^2 = 4a(y - 4)
Now, let's substitute the given equation (y = -(x + 3)^2 + 4) into this form:
(x + 3)^2 = 4a(-(x + 3)^2 + 4 - 4)
(x + 3)^2 = 4a(-(x + 3)^2)
Next, let's expand and simplify:
(x + 3)^2 = -4a(x^2 + 6x + 9)
x^2 + 6x + 9 = -4ax^2 - 24ax - 36a
Now, we can gather like terms and put the equation in the standard form:
5ax^2 + (24a + 6)x + 36a + 9 = 0
Since we have a quadratic equation, we can compare the coefficients of the standard form (Ax^2 + Bx + C) to determine the value of 'a'. Comparing the quadratic equation with the standard form, we get:
A = 5a
B = (24a + 6)
C = 36a + 9
Comparing B with 2√(AC), we can solve for 'a':
B = 2√(AC)
(24a + 6) = 2√((5a)(36a + 9))
(24a + 6) = 2√(180a^2 + 45a)
12a + 3 = √(180a^2 + 45a)
(12a + 3)^2 = 180a^2 + 45a
144a^2 + 72a + 9 = 180a^2 + 45a
-36a^2 + 27a + 9 = 0
-4a^2 + 3a + 1 = 0
(a - 1)(4a + 1) = 0
Solving the quadratic equation for 'a', we find two solutions:
a = 1 or a = -1/4
Therefore, the equation of the parabola can be written as follows:
1. For a = 1: (x + 3)^2 = 4(y - 4)
2. For a = -1/4: (x + 3)^2 = -y + 4
Therefore, the parabola equation in the form "y^2 = 4ax" is (x + 3)^2 = 4(y - 4), and the parabola equation in the form "x^2 = 4ay" is (x + 3)^2 = -y + 4.