To find the x-intercepts of the parabola with a vertex at (1, 9) and a y-intercept at (0, 6), you’ll need to start by using the general form of a parabola’s equation. The vertex form of a parabola is given by:
y = a(x – h)² + k
Here, (h, k) is the vertex of the parabola. For this case, plugging in the vertex (1, 9), we have:
y = a(x – 1)² + 9
Next, we can use the given y-intercept at (0, 6) to find the coefficient ‘a’. By substituting x = 0 and y = 6 into the equation:
6 = a(0 – 1)² + 9
Now, simplify:
6 = a(1) + 9
6 = a + 9
Subtracting 9 from both sides gives:
a = 6 – 9
a = -3
Now we have the complete equation of the parabola:
y = -3(x – 1)² + 9
To find the x-intercepts, we need to set y to 0:
0 = -3(x – 1)² + 9
Rearranging gives us:
3(x – 1)² = 9
Dividing both sides by 3:
(x – 1)² = 3
Next, take the square root of both sides:
x – 1 = ±√3
Solving for x gives:
x = 1 + √3 or x = 1 – √3
Thus, the x-intercepts of the parabola are:
(1 + √3, 0) and (1 – √3, 0).
In numerical form, these approximate to:
(2.73, 0) and (-0.73, 0).
So, to summarize, the x-intercepts of the parabola with the given vertex and y-intercept are (1 + √3, 0) and (1 – √3, 0).