Answer:
The coordinates of the vertex is (1, -6).
Step-by-step explanation:
Given the parabola, [tex]y = \frac{1}{2}(x - 1)^{2} - 6[/tex], where a = [tex]\frac{1}{2}[/tex], and the vertex is represented by (h, k) = (1, -6). Since it is an upward-facing parabola, the vertex is the minimum point in the graph.
I came up with the formula for the quadratic equation by using the coordinates for the vertex (1, -6) and the y-intercept, (0, -5.5) from the given graph. The y-intercept is the point in the graph where the parabola crosses the y-axis, and the value of its x-coordinate is 0.
I Plug the following values into the quadratic equation in vertex form: [tex]y = a(x - h)^{2} + k[/tex].
Let y = -5.5
x = 0
h = 1
k = -6
Since we need to find the value of a:
[tex]y = a(x - h)^{2} + k[/tex]
[tex]-5.5 = a(0 - 1)^{2} - 6[/tex]
[tex]-5.5 = a(- 1)^{2} - 6[/tex]
[tex]-5.5 = 1a - 6[/tex]
Add 6 to both sides to isolate 1a:
-5.5 + 6 = 1a - 6 + 6
0.5 = 1a
Divide both sides by 1 to solve for a:
[tex]\frac{0.5}{1} = \frac{1a}{1}[/tex]
0.5 or 1/2 = a
The value of a determines whether the graph opens up or down. Since the value of a is positive, the graph of the parabola opens upward. Therefore, the formula of the parabola in vertex form is:
[tex]y = \frac{1}{2}(x - 1)^{2} - 6[/tex]
