Question 128198
{{{f(x) = x^2-6x +8}}} Start with the given function



To find the x-intercept, let {{{f(x)=0}}}


{{{0= x^2-6x +8}}} Plug in {{{f(x)=0}}}




{{{0=(x-4)(x-2)}}} Factor the right 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-4=0}}} or  {{{x-2=0}}} 


{{{x=4}}} or  {{{x=2}}}    Now solve for x in each case




So the x-intercepts are (2,0) and (4,0)




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



{{{f(x) = x^2-6x +8}}} Start with the given function



{{{f(0)=1(0)^2-6(0)+8}}} Plug in {{{x=0}}}



{{{f(0)=1*0-6*0+8}}} Raise 0 to the 2nd power to get 0



{{{f(0)=0-6*0+8}}} Multiply 1 and 0 to get 0



{{{f(0)=0-0+8}}} Multiply 6 and 0 to get 0



{{{f(0)=0+8}}} Subtract 0 from 0 to get 0



{{{f(0)=8}}} Add 0 and 8 to get 8



So when {{{x=0}}}, we have {{{y=8}}}




So this means that the y-intercept is (0,8)






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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:


{{{x=-b/(2a)}}}


From the equation {{{y=x^2-6x+8}}} we can see that a=1 and b=-6


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



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



{{{x=(6)/2}}} Multiply 2 and 1 to get 2




{{{x=3}}} Reduce



So the axis of symmetry is  {{{x=3}}}



So the x-coordinate of the vertex is {{{x=3}}}. Lets plug this into the equation to find the y-coordinate of the vertex.



Lets evaluate {{{f(3)}}}


{{{f(x)=x^2-6x+8}}} Start with the given polynomial



{{{f(3)=(3)^2-6(3)+8}}} Plug in {{{x=3}}}



{{{f(3)=(9)-6(3)+8}}} Raise 3 to the second power to get 9



{{{f(3)=(9)-18+8}}} Multiply 6 by 3 to get 18



{{{f(3)=-1}}} Now combine like terms



So the vertex is (3,-1)





So putting all of this together, we get



{{{ graph( 500, 500, -10, 10, -10, 10, x^2-6x+8) }}} Graph of {{{f(x)=x^2-6x+8}}} 



and we can see that the x-intercepts are (2,0) and (4,0), the graph has the y-intercept (0,8), and the vertex is (3,-1). So this visually verifies our answer.