SOLUTION: Find the vertex and intercepts for h(x)= -x^2-2x+8 and sketch the graph?

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Vertex:

To find the vertex, we first need to find the axis of symmetry (ie the x-coordinate of the vertex)
To find the axis of symmetry, use this formula:



From the equation we can see that a=-1 and b=-2

Plug in b=-2 and a=-1


Negate -2 to get 2


Multiply 2 and -1 to get -2



Reduce


So the axis of symmetry is


So the x-coordinate of the vertex is . Lets plug this into the equation to find the y-coordinate of the vertex.


Lets evaluate

Start with the given polynomial


Plug in


Raise -1 to the second power to get 1


Multiply 2 by -1 to get -2


Negate any negatives


Now combine like terms


So the vertex is (-1,9)

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Intercepts:

Y-intercept:

To find the y-intercept, simply plug in x=0 and simplify



Start with the given function


Plug in


Raise 0 to the 2nd power to get 0


Multiply -1 and 0 to get 0


Multiply 2 and 0 to get 0


Subtract 0 from 0 to get 0


Add 0 and 8 to get 8


So when , we have which means that the y-intercept is (0,8)




X-intercept:

To find the x-intercept, plug in y=0 and solve for x


Start with the given equation

Factor the left side (note: if you need help with factoring, check out this solver)



Now set each factor equal to zero:
or

or Now solve for x in each case


So our answer is
or


which means that the x-intercepts are

(-4,0) and (2,0)



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Notice if we graph , we can visually verify our answer