To find the equation of a parabola with the given conditions, we first note that the information provided includes:
- The axis of symmetry is at x = 8.
- The maximum height (vertex) is at y = 1.
- The parabola passes through the point (9, 1).
The standard form of the equation of a parabola with a vertical axis of symmetry is:
y = a(x – h)² + k
In this equation:
- (h, k) is the vertex of the parabola.
- ‘a’ determines the width and direction of the parabola (upward if a is positive, downward if a is negative).
Given that the axis of symmetry is at x = 8 and the maximum height is 1, we can define our vertex:
h = 8 and k = 1. Therefore, our equation becomes:
y = a(x – 8)² + 1
Next, we use the point (9, 1) to find the value of ‘a’. Plugging in x = 9 and y = 1:
1 = a(9 – 8)² + 1
1 = a(1)² + 1
1 = a + 1
This simplifies to:
a = 0.
This indicates that our equation does not actually represent a traditional parabola but rather a horizontal line at y = 1. This situation arises because the maximum height is equal to the height at the point (9,1) you provided. Thus, while the axis of symmetry and other parameters were helpful, they do not describe a parabola in this instance.
Finally, the equation of the function that describes the parabola given these conditions may not exist in a conventional sense, as we’ve established that no actual parabola forms under the given parameters. Therefore, any image of a parabola is flat at that maximum height.