To find the x-intercepts of a parabola, we first need to establish its equation based on the given vertex and y-intercept. The vertex form of a parabola is given by:
y = a(x – h)² + k
Here, (h, k) is the vertex of the parabola. From the question, we know that the vertex is (1, 9). Thus, we can write:
y = a(x – 1)² + 9
Next, we can use the y-intercept to find the value of ‘a’. The y-intercept occurs when x = 0 and is given as (0, 6). Substituting these values into the equation gives:
6 = a(0 – 1)² + 9
This simplifies to:
6 = a(1) + 9
Then, solving for ‘a’:
6 = a + 9
a = 6 – 9
a = -3
Now we can write the complete equation of the parabola:
y = -3(x – 1)² + 9
To find the x-intercepts, we set y = 0 and solve for x:
0 = -3(x – 1)² + 9
This can be rearranged as:
3(x – 1)² = 9
Dividing both sides by 3 gives:
(x – 1)² = 3
Taking the square root of both sides results in:
x – 1 = ±√3
Finally, solving for x, we get:
x = 1 ± √3
Thus, the x-intercepts of the parabola are:
- x = 1 + √3
- x = 1 – √3