Suppose a parabola has an axis of symmetry at x = −2, a maximum height of 7 and also passes through the point (−1, 6). What is the equation of the parabola in vertex form?
To find the equation of the parabola in vertex form, we can use the vertex form equation:
y = a(x - h)^2 + k
where (h, k) represents the vertex of the parabola.
Given that the parabola has an axis of symmetry at x = -2, we can conclude that the vertex lies on this line. So the x-coordinate of the vertex is -2.
The maximum height of the parabola is 7, which means the y-coordinate of the vertex is 7.
Therefore, the vertex is (-2, 7).
Plugging these values into the vertex form equation, we get:
y = a(x - (-2))^2 + 7
y = a(x + 2)^2 + 7
Now, we need to find the value of 'a'.
Since the parabola passes through the point (-1, 6), we can substitute these values into the equation:
6 = a(-1 + 2)^2 + 7
6 = a(1)^2 + 7
6 = a + 7
a = -1
Therefore, the equation of the parabola in vertex form is:
y = -1(x + 2)^2 + 7
To find the equation of a parabola in vertex form, we need the vertex coordinates and one other point on the parabola.
Given that the axis of symmetry is at x = -2, we know that the x-coordinate of the vertex is -2.
To find the y-coordinate of the vertex, we can use the fact that the parabola has a maximum height of 7. Since the vertex of a parabola with a maximum is at the highest point, the y-coordinate of the vertex will be the maximum height, which is 7.
So the vertex coordinates are (-2, 7).
Now, we have the vertex coordinates (-2, 7) and the point (-1, 6). We can use these to write the equation of the parabola in vertex form, which is given by:
y = a(x - h)^2 + k
Where (h, k) denotes the vertex coordinates.
Replacing h with -2, k with 7, x with -1, and y with 6, we get:
6 = a(-1 - (-2))^2 + 7
Simplifying:
6 = a(1)^2 + 7
6 = a + 7
Subtracting 7 from both sides:
-1 = a
Therefore, the value of a is -1.
Finally, substituting the value of a into the equation, we get the equation of the parabola in vertex form:
y = -1(x + 2)^2 + 7
So the equation of the parabola in vertex form is y = -(x + 2)^2 + 7.
from the first sentence, we know
y = a(x+2)^2 + 7
now, use y(-1)=6 to find the value of a.