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
show the steps
To write the equation of a parabola in vertex form, we will use the general equation:
y = a(x - h)^2 + k
where (h, k) represents the coordinates of the vertex.
Given that the vertex of the parabola is (6, -6), we have h = 6 and k = -6.
We also know that the parabola passes through the point (8, -14). We can substitute these values into the equation:
-14 = a(8 - 6)^2 - 6
Simplifying, we get:
-14 = a(2)^2 - 6
-14 = 4a - 6
Now, we can solve for 'a':
4a = -14 + 6
4a = -8
a = -2
Substituting the value of 'a' back into the equation, we can write the equation of the parabola in vertex form:
y = -2(x - 6)^2 - 6
So, 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 will use the standard vertex form equation for a parabola, which is:
y = a(x - h)^2 + k
Where (h, k) represents the vertex of the parabola and a represents the coefficient that determines the shape and direction of the parabola.
Given that the vertex of the parabola is (6, -6), we can substitute these values into the equation:
y = a(x - 6)^2 - 6
Now, we need to find the value of a. To do this, we can use the fact that the parabola passes through the point (8, -14). By substituting these coordinates into the equation, we can solve for a.
-14 = a(8 - 6)^2 - 6
-14 = a(2)^2 - 6
-14 = 4a - 6
4a = -14 + 6
4a = -8
a = -8/4
a = -2
Now that we have the value of a, we can substitute it back into the equation to get the final equation of the parabola:
y = -2(x - 6)^2 - 6