Function 1
![f(x)=- x^{2} +4x+8](https://tex.z-dn.net/?f=f%28x%29%3D-%20x%5E%7B2%7D%20%2B4x%2B8)
First step: Finding when
![f(x)](https://tex.z-dn.net/?f=f%28x%29)
is minimum/maximum
The function has a negative value
![x^{2}](https://tex.z-dn.net/?f=%20x%5E%7B2%7D%20)
hence the
![f(x)](https://tex.z-dn.net/?f=f%28x%29)
has a maximum value which happens when
![x=- \frac{b}{2a}=- \frac{4}{(2)(1)}=2](https://tex.z-dn.net/?f=x%3D-%20%5Cfrac%7Bb%7D%7B2a%7D%3D-%20%5Cfrac%7B4%7D%7B%282%29%281%29%7D%3D2)
. The foci of this parabola lies on
![x=2](https://tex.z-dn.net/?f=x%3D2)
.
Second step: Find the value of y-coordinate by substituting
![x=2](https://tex.z-dn.net/?f=x%3D2)
into
![f(x)](https://tex.z-dn.net/?f=f%28x%29)
which give
![y=- (2)^{2} +4(2)+8=12](https://tex.z-dn.net/?f=y%3D-%20%282%29%5E%7B2%7D%20%2B4%282%29%2B8%3D12)
Third step: Find the distance of the foci from the y-coordinate
![y=- x^{2} +4x+8](https://tex.z-dn.net/?f=y%3D-%20x%5E%7B2%7D%20%2B4x%2B8)
- Multiply all term by -1 to get a positive
![x^{2}](https://tex.z-dn.net/?f=%20x%5E%7B2%7D%20)
![-y= x^{2} -4x-8](https://tex.z-dn.net/?f=-y%3D%20x%5E%7B2%7D%20-4x-8)
- then manipulate the constant of y to get a multiply of 4
![4(- \frac{1}{4})y= x^{2} -4x-8](https://tex.z-dn.net/?f=4%28-%20%5Cfrac%7B1%7D%7B4%7D%29y%3D%20x%5E%7B2%7D%20-4x-8%20)
So the distance of focus is 0.25 to the south of y-coordinates of the maximum, which is
![12- \frac{1}{4}=11.75](https://tex.z-dn.net/?f=12-%20%5Cfrac%7B1%7D%7B4%7D%3D11.75%20)
Hence the coordinate of the foci is (2, 11.75)
Function 2:
![f(x)= 2x^{2}+16x+18](https://tex.z-dn.net/?f=f%28x%29%3D%202x%5E%7B2%7D%2B16x%2B18)
The function has a positive
![x^{2}](https://tex.z-dn.net/?f=%20x%5E%7B2%7D%20)
so it has a minimum
First step -
![x=- \frac{b}{2a}=- \frac{16}{(2)(2)}=-4](https://tex.z-dn.net/?f=x%3D-%20%5Cfrac%7Bb%7D%7B2a%7D%3D-%20%5Cfrac%7B16%7D%7B%282%29%282%29%7D%3D-4%20%20)
Second step -
![y=2(-4)^{2}+16(-4)+18=-14](https://tex.z-dn.net/?f=y%3D2%28-4%29%5E%7B2%7D%2B16%28-4%29%2B18%3D-14%20)
Third step - Manipulating
![f(x)](https://tex.z-dn.net/?f=f%28x%29)
to leave
![x^{2}](https://tex.z-dn.net/?f=%20x%5E%7B2%7D%20)
with constant of 1
![y=2 x^{2} +16x+18](https://tex.z-dn.net/?f=y%3D2%20x%5E%7B2%7D%20%2B16x%2B18)
- Divide all terms by 2
![\frac{1}{2}y= x^{2} +8x+9](https://tex.z-dn.net/?f=%20%5Cfrac%7B1%7D%7B2%7Dy%3D%20x%5E%7B2%7D%20%2B8x%2B9%20)
- Manipulate the constant of y to get a multiply of 4
So the distance of focus from y-coordinate is
![\frac{1}{8}](https://tex.z-dn.net/?f=%20%5Cfrac%7B1%7D%7B8%7D%20)
to the north of
![y=-14](https://tex.z-dn.net/?f=y%3D-14)
Hence the coordinate of foci is (-4, -14+0.125) = (-4, -13.875)
Function 3:
![f(x)=-2 x^{2} +5x+14](https://tex.z-dn.net/?f=f%28x%29%3D-2%20x%5E%7B2%7D%20%2B5x%2B14)
First step: the function's maximum value happens when
![x=- \frac{b}{2a}=- \frac{5}{(-2)(2)}= \frac{5}{4}=1.25](https://tex.z-dn.net/?f=x%3D-%20%5Cfrac%7Bb%7D%7B2a%7D%3D-%20%5Cfrac%7B5%7D%7B%28-2%29%282%29%7D%3D%20%5Cfrac%7B5%7D%7B4%7D%3D1.25%20%20%20)
Second step:
![y=-2(1.25)^{2}+5(1.25)+14=17.125](https://tex.z-dn.net/?f=y%3D-2%281.25%29%5E%7B2%7D%2B5%281.25%29%2B14%3D17.125%20)
Third step: Manipulating
![f(x)](https://tex.z-dn.net/?f=f%28x%29)
![y=-2 x^{2} +5x+14](https://tex.z-dn.net/?f=y%3D-2%20x%5E%7B2%7D%20%2B5x%2B14)
- Divide all terms by -2
![-2y= x^{2} -2.5x-7](https://tex.z-dn.net/?f=-2y%3D%20x%5E%7B2%7D%20-2.5x-7)
- Manipulate coefficient of y to get a multiply of 4
![4(- \frac{1}{8})y= x^{2} -2.5x-7](https://tex.z-dn.net/?f=4%28-%20%5Cfrac%7B1%7D%7B8%7D%29y%3D%20x%5E%7B2%7D%20-2.5x-7%20)
So the distance of the foci from the y-coordinate is -
![\frac{1}{8}](https://tex.z-dn.net/?f=%20%5Cfrac%7B1%7D%7B8%7D%20)
south to y-coordinate
Hence the coordinate of foci is (1.25, 17)
Function 4: following the steps above, the maximum value is when
![x=8.5](https://tex.z-dn.net/?f=x%3D8.5)
and
![y=79.25](https://tex.z-dn.net/?f=y%3D79.25)
. The distance from y-coordinate is 0.25 to the south of y-coordinate, hence the coordinate of foci is (8.5, 79.25-0.25)=(8.5,79)
Function 5: the minimum value of the function is when
![x=-2.75](https://tex.z-dn.net/?f=x%3D-2.75)
and
![y=-10.125](https://tex.z-dn.net/?f=y%3D-10.125)
. Manipulating coefficient of y, the distance of foci from y-coordinate is
![\frac{1}{8}](https://tex.z-dn.net/?f=%20%5Cfrac%7B1%7D%7B8%7D%20)
to the north. Hence the coordinate of the foci is (-2.75, -10.125+0.125)=(-2.75, -10)
Function 6: The maximum value happens when
![x=1.5](https://tex.z-dn.net/?f=x%3D1.5)
and
![y=9.5](https://tex.z-dn.net/?f=y%3D9.5)
. The distance of the foci from the y-coordinate is
![\frac{1}{8}](https://tex.z-dn.net/?f=%20%5Cfrac%7B1%7D%7B8%7D%20)
to the south. Hence the coordinate of foci is (1.5, 9.5-0.125)=(1.5, 9.375)