How do you tell whether the graph opens up or down, find the vertex, and find the axis of symmetry given #y=-x^2-2x-1#?

Question

Eva Adamson · Accepted Answer

The graph opens down.
The vertex is #(-1, 2)#.
The axis of symmetry is #x = -1#.

The graph opens down because the quadratic slope is negative. #(-x) -># slope is #-1# . This equation is in standard form, meaning that it has the form #ax^2 + bx + c#. To find the vertex, we have to use #-b/(2a)# to find the x-coordinate of the vertex. So we know a is -1 and b is -2. Let's put that into the formula #-b/(2a)#. #(-(-2))/(2(-1))# #2/-2# #-1# The x-coordinate of the vertex is #-1#. Now, in order to find the y-coordinate of the vertex, we we plug the x-coordinate value of the vertex back into the equation where x is: #y_v = -(-1) - 2(-1) -1# (#y_v# means y value of vertex) #y_v = 1 + 2 - 1# #y_v = 2# So the coordinate of the vertex is #(-1, 2)#. Now, the axis of symmetry means where the graph reflects. So we know that the vertex is #(-1, 2)#. It becomes symmetrical at the line of #x = -1#, or the line of the x-axis at the vertex.

Eli Assouline · Answer

undefined To determine whether the graph opens up or down:
- Check the coefficient of the x^2 term.
- If the coefficient is negative, the graph opens downward. If it's positive, the graph opens upward.

To find the vertex:
- Use the formula for the x-coordinate of the vertex: x = -b / (2a)
- Substitute the values of a and b from the equation y = ax^2 + bx + c into the formula to find the x-coordinate.
- Once you have the x-coordinate, substitute it back into the original equation to find the y-coordinate.

To find the axis of symmetry:
- The axis of symmetry is a vertical line passing through the vertex.
- Use the x-coordinate of the vertex to determine the equation of the axis of symmetry.