Question 147564
<font size=4><b>Vertex:</b></font>



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=(-b)/(2a)}}}.



{{{x=(-b)/(2a)}}} Start with the given formula.



From {{{y=-x^2-6x+1}}}, we can see that {{{a=-1}}}, {{{b=-6}}}, and {{{c=1}}}.



{{{x=(-(-6))/(2(-1))}}} Plug in {{{a=-1}}} and {{{b=-6}}}.



{{{x=(6)/(2(-1))}}} Negate {{{-6}}} to get {{{6}}}.



{{{x=(6)/(-2)}}} 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^2-6x+1}}} Start with the given equation.



{{{y=-(-3)^2-6(-3)+1}}} Plug in {{{x=-3}}}.



{{{y=-x^2-6x+1}}} Start with the given equation.



{{{y=-(-3)^2-6(-3)+1}}} Plug in {{{x=-3}}}.



{{{y=-1(9)-6(-3)+1}}} Square {{{-3}}} to get {{{9}}}.



{{{y=-9-6(-3)+1}}} Multiply {{{-1}}} and {{{9}}} to get {{{-9}}}.



{{{y=-9+18+1}}} 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 *[Tex \LARGE \left(-3,10\right)].



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<font size=4><b>Intercepts:</b></font>


Y-intercept:


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



{{{y=-x^2-6x+1}}} Start with the given equation.



{{{y=-(0)^2-6(0)+1}}} Plug in {{{x=0}}}.



{{{y=-1(0)-6(0)+1}}} Square {{{0}}} to get {{{0}}}.



{{{y=0-6(0)+1}}} Multiply {{{-1}}} and {{{0}}} to get {{{0}}}.



{{{y=0+0+1}}} 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).



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X-intercept:



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



{{{y=-x^2-6x+1}}} Start with the given equation.



{{{0=-x^2-6x+1}}} Plug in {{{y=0}}}



Let's use the quadratic formula to solve for x



{{{x = (-b +- sqrt( b^2-4ac ))/(2a)}}} Start with the quadratic formula



{{{x = (-(-6) +- sqrt( (-6)^2-4(-1)(1) ))/(2(-1))}}} Plug in  {{{a=-1}}}, {{{b=-6}}}, and {{{c=1}}}



{{{x = (6 +- sqrt( (-6)^2-4(-1)(1) ))/(2(-1))}}} Negate {{{-6}}} to get {{{6}}}. 



{{{x = (6 +- sqrt( 36-4(-1)(1) ))/(2(-1))}}} Square {{{-6}}} to get {{{36}}}. 



{{{x = (6 +- sqrt( 36--4 ))/(2(-1))}}} Multiply {{{4(-1)(1)}}} to get {{{-4}}}



{{{x = (6 +- sqrt( 36+4 ))/(2(-1))}}} Rewrite {{{sqrt(36--4)}}} as {{{sqrt(36+4)}}}



{{{x = (6 +- sqrt( 40 ))/(2(-1))}}} Add {{{36}}} to {{{4}}} to get {{{40}}}



{{{x = (6 +- sqrt( 40 ))/(-2)}}} Multiply {{{2}}} and {{{-1}}} to get {{{-2}}}. 



{{{x = (6 +- 2*sqrt(10))/(-2)}}} Simplify the square root  (note: If you need help with simplifying square roots, check out this <a href=http://www.algebra.com/algebra/homework/Radicals/simplifying-square-roots.solver> solver</a>)  



{{{x = (6)/(-2) +- (2*sqrt(10))/(-2)}}} Break up the fraction.  



{{{x = -3 +- -1*sqrt(10)}}} Reduce.  




{{{x = -3-sqrt(10)}}} or {{{x = -3+sqrt(10)}}}  Break up the expression.  



So our answers are {{{x = -3-sqrt(10)}}} or {{{x = -3+sqrt(10)}}} 



which approximate to {{{x=-6.162}}} or {{{x=0.162}}} 



So the x-intercepts are roughly *[Tex \LARGE \left(-6.162,0\right)] and *[Tex \LARGE \left(0.162,0\right)]




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



{{{ graph( 500, 500, -10, 10, -10, 10, -x^2-6x+1) }}} Graph of {{{y=-x^2-6x+1}}}