SOLUTION: What does it mean to find the vertex, intercepts, and sketch of y=-x2-6x+1?

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Question 147564: What does it mean to find the vertex, intercepts, and sketch of y=-x2-6x+1?
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
Vertex:


In order to find the vertex, we first need to find the x-coordinate of the vertex.


To find the x-coordinate of the vertex, use this formula: x=%28-b%29%2F%282a%29.


x=%28-b%29%2F%282a%29 Start with the given formula.


From y=-x%5E2-6x%2B1, we can see that a=-1, b=-6, and c=1.


x=%28-%28-6%29%29%2F%282%28-1%29%29 Plug in a=-1 and b=-6.


x=%286%29%2F%282%28-1%29%29 Negate -6 to get 6.


x=%286%29%2F%28-2%29 Multiply 2 and -1 to get -2.


x=-3 Divide.


So the x-coordinate of the vertex is x=-3. Note: this means that the axis of symmetry is also x=-3.


Now that we know the x-coordinate of the vertex, we can use it to find the y-coordinate of the vertex.


y=-x%5E2-6x%2B1 Start with the given equation.


y=-%28-3%29%5E2-6%28-3%29%2B1 Plug in x=-3.


y=-x%5E2-6x%2B1 Start with the given equation.


y=-%28-3%29%5E2-6%28-3%29%2B1 Plug in x=-3.


y=-1%289%29-6%28-3%29%2B1 Square -3 to get 9.


y=-9-6%28-3%29%2B1 Multiply -1 and 9 to get -9.


y=-9%2B18%2B1 Multiply -6 and -3 to get 18.


y=10 Combine like terms.


So the y-coordinate of the vertex is y=10.


So the vertex is .





Intercepts:

Y-intercept:

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


y=-x%5E2-6x%2B1 Start with the given equation.


y=-%280%29%5E2-6%280%29%2B1 Plug in x=0.


y=-1%280%29-6%280%29%2B1 Square 0 to get 0.


y=0-6%280%29%2B1 Multiply -1 and 0 to get 0.


y=0%2B0%2B1 Multiply -6 and 0 to get 0.


y=1 Combine like terms.

So when x=0, y=1. This means that the y-intercept is (0,1).


------------------

X-intercept:


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


y=-x%5E2-6x%2B1 Start with the given equation.


0=-x%5E2-6x%2B1 Plug in y=0


Let's use the quadratic formula to solve for x


x+=+%28-b+%2B-+sqrt%28+b%5E2-4ac+%29%29%2F%282a%29 Start with the quadratic formula


x+=+%28-%28-6%29+%2B-+sqrt%28+%28-6%29%5E2-4%28-1%29%281%29+%29%29%2F%282%28-1%29%29 Plug in a=-1, b=-6, and c=1


x+=+%286+%2B-+sqrt%28+%28-6%29%5E2-4%28-1%29%281%29+%29%29%2F%282%28-1%29%29 Negate -6 to get 6.


x+=+%286+%2B-+sqrt%28+36-4%28-1%29%281%29+%29%29%2F%282%28-1%29%29 Square -6 to get 36.


x+=+%286+%2B-+sqrt%28+36--4+%29%29%2F%282%28-1%29%29 Multiply 4%28-1%29%281%29 to get -4


x+=+%286+%2B-+sqrt%28+36%2B4+%29%29%2F%282%28-1%29%29 Rewrite sqrt%2836--4%29 as sqrt%2836%2B4%29


x+=+%286+%2B-+sqrt%28+40+%29%29%2F%282%28-1%29%29 Add 36 to 4 to get 40


x+=+%286+%2B-+sqrt%28+40+%29%29%2F%28-2%29 Multiply 2 and -1 to get -2.


x+=+%286+%2B-+2%2Asqrt%2810%29%29%2F%28-2%29 Simplify the square root (note: If you need help with simplifying square roots, check out this solver)


x+=+%286%29%2F%28-2%29+%2B-+%282%2Asqrt%2810%29%29%2F%28-2%29 Break up the fraction.


x+=+-3+%2B-+-1%2Asqrt%2810%29 Reduce.



x+=+-3-sqrt%2810%29 or x+=+-3%2Bsqrt%2810%29 Break up the expression.


So our answers are x+=+-3-sqrt%2810%29 or x+=+-3%2Bsqrt%2810%29


which approximate to x=-6.162 or x=0.162


So the x-intercepts are roughly and






With all of this info, we can now sketch the graph:


+graph%28+500%2C+500%2C+-10%2C+10%2C+-10%2C+10%2C+-x%5E2-6x%2B1%29+ Graph of y=-x%5E2-6x%2B1