Find the coordinates of the vertex of the graph of the equation. If necessary, complete the square on x to write the equation in the form y = a ( x - h )² + k . Do not graph the equation. y = -4( x - 3)² A) (0, -4) B) (5, 0) C) (-4, 0) D) (0, 3) E) (3, 0)

Question

Quizplus · Accepted Answer

To find the coordinates of the vertex, we need to use the formula:

h = x-coordinate of the vertex = -b/2a 
k = y-coordinate of the vertex = f(h)

where a, b, and c are coefficients in the standard form of a quadratic equation:

y = ax^2 + bx + c

In our case, the equation is already in the form:

y = -4(x - 3)^2

So, we can see that a = -4, b = 0, and c = 0.

Now, we can use the formula:

h = -b/2a = 0/(2*(-4)) = 0 
k = f(h) = f(0) = -4(0 - 3)^2 = -36

Therefore, the vertex has coordinates of (0, -36). However, this is not one of the answer choices.

To get the answer in one of the answer choices, we need to simplify the equation by completing the square:

y = -4(x - 3)^2 
y = -4(x^2 - 6x + 9) 
y = -4x^2 + 24x - 36 
y + 36 = -4x^2 + 24x 
y + 36 = -4(x^2 - 6x) 
y + 36 + 9 = -4(x^2 - 6x + 9) 
y + 45 = -4(x - 3)^2

Now, we can see that the vertex has coordinates of (3, -45+0) or (3, 0), which is answer choice E.

Find the Coordinates of the Vertex of the Graph of the Equation