Question 486972


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=(1/3)x^2+2x+1}}}, we can see that {{{a=1/3}}}, {{{b=2}}}, and {{{c=1}}}.



{{{x=(-(2))/(2(1/3))}}} Plug in {{{a=1/3}}} and {{{b=2}}}.



{{{x=(-2)/(2/3)}}} Multiply 2 and {{{1/3}}} to get {{{2/3}}}.



{{{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.



{{{f(x)=(1/3)x^2+2x+1}}} Start with the given equation.



{{{f(-3)=(1/3)(-3)^2+2(-3)+1}}} Plug in {{{x=-3}}}.



{{{f(-3)=(1/3)(9)+2(-3)+1}}} Square {{{-3}}} to get {{{9}}}.



{{{f(-3)=3+2(-3)+1}}} Multiply {{{1/3}}} and {{{9}}} to get {{{3}}}.



{{{f(-3)=3-6+1}}} Multiply {{{2}}} and {{{-3}}} to get {{{-6}}}.



{{{f(-3)=-2}}} Combine like terms.



So the y-coordinate of the vertex is {{{y=-2}}}.



So the vertex is *[Tex \LARGE \left(-3,-2\right)].