Suppose a parabola has an axis of symmetry at x = -2, a minimum height at -6, and passes through the point (0, 10). Write the equation of the parabola in vertex form.
from the axis and vertex, we know
y = a(x+2) - 6
Now use the point to solve for a. You already know that a is positive (why?)
To write the equation of a parabola in vertex form, we need to use the vertex form equation:
y = a(x-h)^2 + k
Where (h, k) represents the coordinates of the vertex of the parabola.
In the given problem, the parabola has an axis of symmetry at x = -2, and a minimum height at -6. This means that the vertex of the parabola is (-2, -6).
So, we can substitute h = -2 and k = -6 in the vertex form equation:
y = a(x - (-2))^2 + (-6)
y = a(x + 2)^2 - 6
Now, let's use the fact that the parabola passes through the point (0, 10) to find the value of 'a'.
Substitute x = 0 and y = 10 into the equation:
10 = a(0 + 2)^2 - 6
10 = a(2)^2 - 6
10 = 4a - 6
16 = 4a
a = 4
Now we have the value of 'a' which is 4. Substitute it back into the equation:
y = 4(x + 2)^2 - 6
So, the equation of the parabola in vertex form is y = 4(x + 2)^2 - 6.
To write the equation of the parabola in vertex form, we need to know the vertex coordinates. We have been given the axis of symmetry as x = -2, which means the x-coordinate of the vertex is -2.
Since the minimum height of the parabola is at -6, the y-coordinate of the vertex is -6.
Therefore, the vertex is (-2, -6).
The vertex form of a parabola is given by the equation:
y = a(x - h)^2 + k,
where (h, k) represents the vertex coordinates.
In this case, h = -2 and k = -6, so substituting these values into the equation, we have:
y = a(x - (-2))^2 + (-6)
Simplifying, we get:
y = a(x + 2)^2 - 6
Now, we can use the point (0, 10) which lies on the parabola to find the value of a.
Substituting x = 0 and y = 10 into the equation, we have:
10 = a(0 + 2)^2 - 6
10 = 4a - 6
Adding 6 to both sides, we get:
16 = 4a
Dividing both sides by 4, we have:
4 = a
Finally, we substitute this value of a back into the equation to get the final equation of the parabola in vertex form:
y = 4(x + 2)^2 - 6