Question 131104
To find the vertex, we first need to find the line of symmetry (ie the x-coordinate of the vertex)

To find the line of symmetry, use this formula:


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


From the equation {{{y=3x^2-24x+51}}} we can see that a=3 and b=-24


{{{x=(--24)/(2*3)}}} Plug in b=-24 and a=3



{{{x=24/(2*3)}}} Negate -24 to get 24



{{{x=(24)/6}}} Multiply 2 and 3 to get 6




{{{x=4}}} Reduce



So the line of symmetry is  {{{x=4}}}



So the x-coordinate of the vertex is {{{x=4}}}. Lets plug this into the equation to find the y-coordinate of the vertex.



Lets evaluate {{{f(4)}}}


{{{f(x)=3x^2-24x+51}}} Start with the given polynomial



{{{f(4)=3(4)^2-24(4)+51}}} Plug in {{{x=4}}}



{{{f(4)=3(16)-24(4)+51}}} Raise 4 to the second power to get 16



{{{f(4)=48-24(4)+51}}} Multiply 3 by 16 to get 48



{{{f(4)=48-96+51}}} Multiply 24 by 4 to get 96



{{{f(4)=3}}} Now combine like terms



So the vertex is (4,3)





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Answer:



So the line of symmetry is {{{x=4}}} and the vertex is (4,3)




If we graph, we can visually verify our answer


{{{drawing(900,900,-10,10,-10,10,
graph(900,900,-10,10,-10,10, 3x^2-24x+51),
line(4,-12,4,12),
circle(4,3,0.05),
circle(4,3,0.08)
)}}} Graph of {{{f(x)=3x^2-24x+51}}} with the line of symmetry {{{x=4}}} and the vertex (4,3)