Question

please help asap!! 50 pts

Answer 1

Choice D!! If you’re confused the vertex is the turning point, which just so happens to be an x intercept. Knowing that you can write your equation. It bounces so it’s (X+1)^2 expand it distribute everything then distribute a negative in last bc it opens down hope this helps let me know if u have questions

Answer 2

1) The axis of symmetry is the vertical line that divides the parabola in half.
On the graph, it's being shown as the reddish-purple line. You can see this line goes through x = -1, making the axis of symmetry: x = -1.

2) The vertex of a parabola is on the axis of symmetry and it's the highest or lowest point on the parabola, depending on whether the parabola opens up (makes a smiley face) or down (makes a frown). 
In the graph, this parabola opens down, so we're looking for a maximum point that lies on the axis of symmetry. This point is the black dot, making the vertex = (-1, 0).

3) Finally, find the formula of the parabola. The easiest way would be to plug coordinates into each of you remaining equations, after eliminating the wrong ones using steps 1 & 2. Answer choice A is wrong because it wrongly says that the vertex is (-1, -1). You're left with:
Choice B: $-2 x^{2} - 2x -1$
Choice C: $- x^{2} -x +2$
Choice D: $- x^{2} -2x -1$

Plug the vertex, (-1, 0) into each equation, with x = -1 and y = 0:
Choice B: 
$y = -2 x^{2} - 2x -1\\ 0 =-2 (-1)^{2} - 2(-1)-1\\ 0 = -2+2 -1\\ 0 \neq -1$
Choice B doesn't work.

Choice C:
$y = - x^{2} -x +2\\ 0 = - (-1)^{2} -(-1) +2\\ 0 = -1 + 1 + 2\\ 0 \neq 2$
Choice C doesn't work.

Choice D: 
$y = - x^{2} -2x -1\\ 0 = -(-1)^2 - 2(-1) - 1\\ 0 = -1 + 2 -1\\ 0 = 0$
Choice D works!

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Answer: D