I know that this is a lot but I really need help!! Hopefully, the reward will help!

A satellite dish has the shape of a parabola, the U-shaped graph of a quadratic function. Suppose an engineer has determined that the shape of one of the satellite dishes offered by the company can be modeled by the quadratic function y = 2/27x^2 - 4/3x, where y is the vertical depth of the satellite dish in inches and x is the horizontal width in inches.

a) Is the vertex of the function a maximum or minimum point, and how can you tell?

b) Find the x-coordinate of the vertex. Show all work leading your answer and write the answer in simplest form.

c) Find the y-coordinate of the vertex. Show all work leading your answer and write the answer in simplest form.

d) Write the vertex as an ordered pair (x, y). What does the vertex represent for this situation? Write 1 -2 sentences to explain your answer.

a) The leading coefficient (the coefficient of the x² term) is positive, so that means the parabola points up. So the vertex is at the bottom of the parabola, making it a minimum.

b) For a parabola y = ax² + bx + c, the x coordinate of the vertex can be found at x = -b / (2a). Here, a = 2/27 and b = -4/3.

x = -(-4/3) / (2 × 2/27)

x = (4/3) / (4/27)

x = (4/3) × (27/4)

x = 9

c) To find the y coordinate of the vertex, we simply evaluate the function at x=9:

y = 2/27 x² − 4/3 x

y = 2/27 (9)² − 4/3 (9)

y = 6 − 12

y = -6

d) The ordered pair is (9, -6). This means that at the lowest point of the dish, the dish is 6 inches deep. Also, since the dish is symmetrical, and the lowest point is 9 inches from the end, then the total width is double that, or 18 inches.