How do you sketch a graph of this function? `f(x)=x^2+8x+19`

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`f(x)=x^2+8x+19`

Note that graph of a quadratic function is a parabola. To plot it, determine its vertex (h,k). To do so, apply the formula:

`h=-b/(2a)`

where a is the coefficient of x^2 and b coefficient of x.

In the function f(x)=x^2+8x+19, the values of a and b are 1 and 8, respectively. Plug-in these value to the formula to get h.

`h=-8/(2*1)=-8/2=-4`

Then, evaluate the function f(x)=x^2+8x+19 when x=h to get the value of k.

`k=f(h)=f(-4)=(-4)^2+8*(-4)+19`

`k=16-32+19=3`

Hence, the vertex of the parabola is (-4, 3).

Next, use additional two points to plot it.  To do so, assign a value of x that is less than h. And solve for the corresponding value of y.

`x=-6` ,

`y=f(-6)=(-6)^2+8*(-6)+19=36-48+19=7`

Also, assign a value of x that is greater than h.

`x=-2` ,

`y=f(-2)=(-2)^2+8*(-2)+19=4-16+19=7`

So the other two points are (-6,7) and (-2,7).

Then, plot the vertex (-3,4) and the two points (-6,7) and (-2,7). Connect them and extend the parabola on both ends.

Hence, the graph of the given function is:

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