Question 143034
a)


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+4x+5}}} we can see that a=1 and b=4


{{{x=(-4)/(2*1)}}} Plug in b=4 and a=1



{{{x=(-4)/2}}} Multiply 2 and 1 to get 2




{{{x=-2}}} Reduce



So the axis of symmetry is  {{{x=-2}}}



So the x-coordinate of the vertex is {{{x=-2}}}. Lets plug this into the equation to find the y-coordinate of the vertex.



Lets evaluate {{{f(-2)}}}


{{{f(x)=x^2+4x+5}}} Start with the given polynomial



{{{f(-2)=(-2)^2+4(-2)+5}}} Plug in {{{x=-2}}}



{{{f(-2)=(4)+4(-2)+5}}} Raise -2 to the second power to get 4



{{{f(-2)=(4)+-8+5}}} Multiply 4 by -2 to get -8



{{{f(-2)=1}}} Now combine like terms



So the vertex is (-2,1)




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b)


Since {{{a>0}}}, this tells us that the parabola is opening upward and that there is a minimum. At the minimum, there is the vertex. So this means that the minimum is {{{y=1}}} which occurs at {{{x=-2}}}



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c)


Because the min is {{{y=1}}}, this means that the range is from  {{{y=1}}} to infinity. So the range is {{{y>=1}}} which in interval notation is: <font size="8">[</font>*[Tex \LARGE 1,\infty]<font size="8">)</font>



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d)


At the axis of symmetry, the graph changes from increasing to decreasing (or vice versa). Also, since the graph opens upward, this means that the interval that the function decreases is: <font size="8">(</font>*[Tex \LARGE \bf{-\infty,-2}]<font size="8">)</font>



Also, the  interval that the function increases is: <font size="8">(</font>*[Tex \LARGE -2,\infty]<font size="8">)</font>



Here is the graph of {{{y=x^2+4x+5}}} to visually verify our answer


{{{ graph( 500, 500, -10, 10, -10, 10, x^2+4x+5) }}}