Question 148000
<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+x-6}}}, we can see that {{{a=1}}}, {{{b=1}}}, and {{{c=-6}}}.



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



{{{x=(-1)/(2)}}} Multiply 2 and {{{1}}} to get {{{2}}}.



So the x-coordinate of the vertex is {{{x=-1/2}}}. Note: this means that the axis of symmetry is also {{{x=-1/2}}}.



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+x-6}}} Start with the given equation.



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



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



{{{y=1/4-1/2-6}}} Square {{{-1/2}}} to get {{{1/4}}}.



{{{y=-25/4}}} Combine like terms.



So the vertex is *[Tex \LARGE \left(-\frac{1}{2},-\frac{25}{4}\right)]



note: you can verify this with a calculator. You can use the "min/max" feature. 


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




x-intercept:



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



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



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



{{{(x+3)(x-2)=0}}} Factor the left side (note: if you need help with factoring, check out this <a href=http://www.algebra.com/algebra/homework/playground/change-this-name4450.solver>solver</a>)




Now set each factor equal to zero:

{{{x+3=0}}} or  {{{x-2=0}}} 


{{{x=-3}}} or  {{{x=2}}}    Now solve for x in each case


So the x-intercepts are (-3,0) and (2,0)



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y-intercept:



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


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



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



{{{y=-6}}} Simplify


So the y-intercept is (0,-6) 



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Notice if we graph {{{y=x^2+x-6}}}  we can visually verify our answers.



{{{ graph(500,500,-10,10,-10,10,0, x^2+x-6) }}} Graph of {{{y=x^2+x-6}}}