Question

find the vertex on the graph of the equation
y=-x^2-4x+7

Answer 1

To find the vertex of the graph of the equation y = -x^2 - 4x + 7, we can rewrite the equation in vertex form: y = a(x - h)^2 + k, where (h, k) represents the vertex.

First, let's complete the square by adding and subtracting the appropriate constant to the equation:
y = -1(x^2 + 4x) + 7

Now, we need to add a constant inside the parentheses that will make the expression a perfect square. To do this, take half of the coefficient of x (which is 4) and square it:
(4/2)^2 = 2^2 = 4

Add 4 inside the parentheses, but to maintain the equality, we need to subtract 4 outside the parentheses:
y = -1(x^2 + 4x + 4 - 4) + 7

Simplify inside the parentheses and combine like terms:
y = -1((x + 2)^2 - 4) + 7
y = -(x + 2)^2 + 4 + 7
y = -(x + 2)^2 + 11

Now that the equation is in vertex form, we can identify the vertex. The vertex is represented by the coordinates (h, k), so in this case, (-2, 11) is the vertex.

Therefore, the vertex of the graph of the equation y = -x^2 - 4x + 7 is (-2, 11).