there are ___ parabolas that have a vertex (-3,-4) and pass through the point (-1,4)
We shall attempt to derive the Eq of an
X-parabola and a Y-parabola using the
given vertex and point.
V(-3,-4), P(-1,4).
Vertex Form: X = a(Y-K)^2 + h.
X = a(4-(-4))^2 -3 = -1,
64a - 3 = -1,
64a = -1 + 3 = 2,
a = 2/64 = 1/32.
Eq: X = (1/32)(Y+4)^2 - 3.
Therefore an X-parabola with the given
vertex passes through the point given.
Y = a(X-h)^2 + K.
Y = a(-1-(-3))^2 - 4 = 4,
4a - 4 = 4,
4a = 8,
a = 2.
Eq: Y = 2(X+3)^2 - 4.
Therefore, a Y-parabola with the given
vertex passes through the givn point.
Answer: 2 parabolas.
To find the number of parabolas that have a given vertex and pass through a specific point, we need to consider the general equation of a parabola in vertex form.
The vertex form of a parabola is given by:
y = a(x-h)^2 + k
where (h, k) represents the vertex of the parabola, and 'a' is a constant that determines the shape and orientation of the parabola.
In this case, we are given the vertex (-3, -4) and a point (-1, 4) that the parabola passes through. Substituting these values into the equation, we get:
4 = a(-1 - (-3))^2 + (-4)
Simplifying further:
4 = a(2)^2 - 4
4 = 4a - 4
8 = 4a
a = 2
So, the value of 'a' in this case is 2.
If we insert this value into the equation, we obtain:
y = 2(x - (-3))^2 + (-4)
y = 2(x + 3)^2 - 4
The equation of the parabola that passes through the given vertex (-3, -4) and the point (-1, 4) is y = 2(x + 3)^2 - 4.
Since there is only one way to determine the equation of a parabola when the vertex and one point that it passes through are given, there is only one parabola that satisfies these conditions.