Write the equation of the parabola that has the vertex at point (3,−12) and passes through the point (0,6).
you know that
y = a(x-3)^2 - 12
so, now just find a by solving
a(0-3)^2 - 12 = 6
To find the equation of the parabola, we need to determine the values of a, b, and c in the standard form equation of a parabola, y = ax^2 + bx + c.
Given that the vertex is (3, -12), we can use this information to find the value of c.
Substituting the vertex coordinates into the equation, we have:
-12 = a(3)^2 + b(3) + c
Simplifying further, we have:
-12 = 9a + 3b + c
Now, we also know that the point (0, 6) lies on the parabola. Substituting these coordinates into the equation, we get:
6 = a(0)^2 + b(0) + c
6 = c
So, now we have the value of c as 6.
Now, substituting this value back into the vertex equation, we have:
-12 = 9a + 3b + 6
Simplifying further, we get:
9a + 3b = -18
Dividing the entire equation by 3, we have:
3a + b = -6
Therefore, the equation of the parabola that satisfies the given conditions is:
y = ax^2 + bx + c
y = ax^2 + bx + 6
And 3a + b = -6.
To write the equation of a parabola given its vertex and another point on the curve, we need to first determine which form of the equation to use.
The vertex form of a parabola's equation is given as follows:
y = a(x - h)^2 + k
Where:
- (h, k) represents the vertex coordinates
- a represents the vertical stretch or compression factor
In this case, the vertex coordinates are (3, -12), so h = 3 and k = -12. Additionally, we have another point (0, 6) that lies on the parabola.
To find the value of a, we can substitute the coordinates of the vertex and the point (0, 6) into the equation and solve for a.
Starting with the vertex form:
y = a(x - 3)^2 - 12
Substituting the coordinates of the point (0, 6):
6 = a(0 - 3)^2 - 12
Simplifying:
6 = 9a - 12
Adding 12 to both sides:
18 = 9a
Dividing both sides by 9:
2 = a
Now that we have the value of a, we can substitute it back into the equation to obtain the final equation of the parabola:
y = 2(x - 3)^2 - 12
Therefore, the equation of the parabola that has the given vertex (3, -12) and passes through the point (0, 6) is y = 2(x - 3)^2 - 12.