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Solnce55 [7]
3 years ago
14

Match the parabolas represented by the equations with their foci.

Mathematics
1 answer:
Elenna [48]3 years ago
8 0

Function 1 f(x)=- x^{2} +4x+8


First step: Finding when f(x) is minimum/maximum
The function has a negative value x^{2} hence the f(x) has a maximum value which happens when x=- \frac{b}{2a}=- \frac{4}{(2)(1)}=2. The foci of this parabola lies on x=2.

Second step: Find the value of y-coordinate by substituting x=2 into f(x) which give y=- (2)^{2} +4(2)+8=12

Third step: Find the distance of the foci from the y-coordinate
y=- x^{2} +4x+8 - Multiply all term by -1 to get a positive x^{2}
-y= x^{2} -4x-8 - then manipulate the constant of y to get a multiply of 4
4(- \frac{1}{4})y= x^{2} -4x-8
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

Hence the coordinate of the foci is (2, 11.75)

Function 2: f(x)= 2x^{2}+16x+18

The function has a positive x^{2} so it has a minimum

First step - x=- \frac{b}{2a}=- \frac{16}{(2)(2)}=-4
Second step - y=2(-4)^{2}+16(-4)+18=-14
Third step - Manipulating f(x) to leave x^{2} with constant of 1
y=2 x^{2} +16x+18 - Divide all terms by 2
\frac{1}{2}y= x^{2} +8x+9 - Manipulate the constant of y to get a multiply of 4
4( \frac{1}{8}y= x^{2} +8x+9

So the distance of focus from y-coordinate is \frac{1}{8} to the north of y=-14
Hence the coordinate of foci is (-4, -14+0.125) = (-4, -13.875)

Function 3: f(x)=-2 x^{2} +5x+14

First step: the function's maximum value happens when x=- \frac{b}{2a}=- \frac{5}{(-2)(2)}= \frac{5}{4}=1.25
Second step: y=-2(1.25)^{2}+5(1.25)+14=17.125
Third step: Manipulating f(x)
y=-2 x^{2} +5x+14 - Divide all terms by -2
-2y= x^{2} -2.5x-7 - Manipulate coefficient of y to get a multiply of 4
4(- \frac{1}{8})y= x^{2} -2.5x-7
So the distance of the foci from the y-coordinate is -\frac{1}{8} 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 and y=79.25. 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 and y=-10.125. Manipulating coefficient of y, the distance of foci from y-coordinate is \frac{1}{8} 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 and y=9.5. The distance of the foci from the y-coordinate is \frac{1}{8} to the south. Hence the coordinate of foci is (1.5, 9.5-0.125)=(1.5, 9.375)

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