Question Suppose a parabola has a vertex (6, -6) and also passes through the point (8, -14). Write the equation of the parabola in vertex form.(1 point) show all your work
The general equation of a parabola in vertex form is y = a(x-h)^2 + k, where (h, k) represents the coordinates of the vertex.
We are given that the vertex is (6, -6). So, h = 6 and k = -6.
We also know that the parabola passes through the point (8, -14). Plugging in these values into the equation, we get:
-14 = a(8-6)^2 + (-6)
Simplifying the equation further:
-14 = a(2)^2 - 6
-14 = 4a - 6
Rearranging the equation to solve for a:
4a = -14 + 6
4a = -8
a = -8/4
a = -2
Now that we have the value of a, we can write the equation of the parabola in vertex form:
y = -2(x-6)^2 - 6
Therefore, the equation of the parabola in vertex form is y = -2(x-6)^2 - 6.
To write the equation of the parabola in vertex form, we can utilize the standard vertex form of a parabola equation:
y = a(x - h)^2 + k
In this equation, (h, k) represents the vertex of the parabola. In our case, the vertex is given as (6, -6). We just need to find the value of 'a' to complete the equation.
To find 'a', we can substitute the coordinates of the given point (8, -14) into the equation and solve for 'a'.
Using the point (8, -14):
-14 = a(8 - 6)^2 + (-6)
Simplifying further:
-14 = 4a - 6
Add 6 to both sides of the equation:
-14 + 6 = 4a
-8 = 4a
Divide both sides of the equation by 4:
-8/4 = a
-2 = a
Now that we have the value of 'a' as -2, we can substitute this value along with the vertex (h, k) into the equation:
y = -2(x - 6)^2 - 6
Therefore, the equation of the parabola in vertex form is y = -2(x - 6)^2 - 6.
To find the equation of a parabola in vertex form, we need to use the formula:
y = a(x - h)^2 + k
Given that the vertex is (6, -6), we can substitute h = 6 and k = -6 into the equation:
y = a(x - 6)^2 - 6
Now, we know that the parabola also passes through the point (8, -14). We can substitute these values into the equation:
-14 = a(8 - 6)^2 - 6
Simplifying further, we get:
-14 = a(2)^2 - 6
-14 = 4a - 6
Next, we can solve for 'a':
-14 + 6 = 4a
-8 = 4a
a = -8/4
a = -2
Now that we have the value of 'a', we can substitute it into the equation:
y = -2(x - 6)^2 - 6
Therefore, the equation of the parabola in vertex form is y = -2(x - 6)^2 - 6.