Given the vertex and a point on the parabola, write an equation in vertex form
Vertex: (-6,-2) ; Point: (-3,4)
The vertex form of a parabola with vertex (h, k) is given by the equation y = a(x - h)^2 + k.
Using the given vertex (-6, -2), we substitute these values into the equation:
y = a(x - (-6))^2 + (-2)
Simplifying:
y = a(x + 6)^2 - 2
Now, we need to find the value of "a" by substituting the coordinates of the given point (-3, 4):
4 = a((-3) + 6)^2 - 2
Simplifying further:
4 = a(3)^2 - 2
4 = 9a - 2
9a = 6
a = 6/9
a = 2/3
Substituting the value of "a" back into the equation, we have:
y = (2/3)(x + 6)^2 - 2
Therefore, the equation in vertex form is y = (2/3)(x + 6)^2 - 2.
To write an equation in vertex form, we can use the formula:
y = a(x - h)^2 + k
where (h, k) represents the vertex of the parabola.
Given the vertex (-6, -2) and a point (-3, 4) on the parabola, we can substitute these values into the equation and solve for 'a.'
Using the vertex coordinates, we have:
-2 = a(-6 - (-6))^2 + (-2)
-2 = a(0)^2 + (-2)
-2 = 0 + (-2)
This simplifies to:
-2 = -2
Since both sides of the equation are equal, 'a' can have any value. Therefore, we can choose any non-zero value for 'a.'
So, the equation in vertex form using the given vertex (-6, -2) is:
y = a(x + 6)^2 - 2
To write the equation of a parabola in vertex form when given the vertex and a point on the parabola, you can use the following steps:
Step 1: Identify the vertex form equation:
The vertex form equation of a parabola is given by:
y = a(x - h)^2 + k,
where (h, k) represents the vertex of the parabola.
Step 2: Substitute the vertex values into the equation:
Given the vertex (-6, -2), we can substitute h = -6 and k = -2 into the equation:
y = a(x - (-6))^2 + (-2),
which simplifies to:
y = a(x + 6)^2 - 2.
Step 3: Substitute the point coordinates into the equation:
Now, using the point (-3, 4), let's substitute x = -3 and y = 4 into the equation:
4 = a((-3) + 6)^2 - 2.
Step 4: Solve for 'a':
Simplifying the equation, we have:
4 = a(3)^2 - 2,
4 = 9a - 2,
6 = 9a,
a = 6/9,
a = 2/3.
Step 5: Substitute 'a' back into the equation:
Now that we have the value of 'a,' we can substitute it back into the equation:
y = (2/3)(x + 6)^2 - 2.
Therefore, the equation in vertex form, given the vertex (-6, -2), and a point (-3, 4), is:
y = (2/3)(x + 6)^2 - 2.