SOLUTION: Based on the equations of the parabolas, identify each parabola whose focus and vertex lie in different quadrants. a. y=-x^2/8+x/4+23/8 b. y=x^2/32+x/4-13/2 c. x=-y^2/16-y/4+1

Algebra ->  Finance -> SOLUTION: Based on the equations of the parabolas, identify each parabola whose focus and vertex lie in different quadrants. a. y=-x^2/8+x/4+23/8 b. y=x^2/32+x/4-13/2 c. x=-y^2/16-y/4+1      Log On


   

Question 1180070: Based on the equations of the parabolas, identify each parabola whose focus and vertex lie in different quadrants.
a. y=-x^2/8+x/4+23/8
b. y=x^2/32+x/4-13/2
c. x=-y^2/16-y/4+11/4
d. x=y^2/16+y/4-19/4
e. x=-y^2/36-5y/18+299/36
f. y=-x^2/24-5x/12+95/24
Found 2 solutions by MathLover1, MathTherapy:
Answer by MathLover1(20850) About Me  (Show Source):
You can put this solution on YOUR website!

Based on the equations of the parabolas, identify each parabola whose focus and vertex lie in different quadrants.

a.
y=-x%5E2%2F8%2Bx%2F4%2B23%2F8
use vertex form
4p%28y-k%29=%28x-h%29%5E2
y=-x%5E2%2F8%2Bx%2F4%2B23%2F8........complete square
y=-%281%2F8%29%28x%5E2-2x%29%2B23%2F8
y=-%281%2F8%29%28x%5E2-2x%2Bb%5E2%29-%28-%281%2F8%29b%5E2%29%2B23%2F8......b=2%2F2=1
y=-%281%2F8%29%28x%5E2-2x%2B1%5E2%29%2B%281%2F8%291%5E2%2B23%2F8
y=-%281%2F8%29%28x-1%29%5E2%2B1%2F8%2B23%2F8
y=-%281%2F8%29%28x-1%29%5E2%2B24%2F8
y=-%281%2F8%29%28x-1%29%5E2%2B3
y-3=-%281%2F8%29%28x-1%29%5E2
%28y-3%29%2F%28-1%2F8%29=%28x-1%29%5E2
-8%28y-3%29=%28x-1%29%5E2

=>h=1, k=3, 4p=-8=>p=-2
vertex is at (h,k)=(1,3)->Q I
focus is at: (h,k%2Bp)=(1,3-2)=(1,1)->Q I
=>focus and vertex lie in same quadrant

b.
y=x%5E2%2F32%2Bx%2F4-13%2F2
y=%281%2F32%29%28x%5E2%2B8x%29-13%2F2
y=%281%2F32%29%28x%5E2%2B8x%2Bb%5E2%29-%281%2F32%29b%5E2-13%2F2.......b=4
y=%281%2F32%29%28x%5E2%2B8x%2B4%5E2%29-%281%2F32%294%5E2-13%2F2
y=%281%2F32%29%28x%2B4%29%5E2-1%2F2-13%2F2
y=%281%2F32%29%28x%2B4%29%5E2-14%2F2
y=%281%2F32%29%28x%2B4%29%5E2-7
y%2B7=%281%2F32%29%28x%2B4%29%5E2
32%28y%2B7%29=%28x%2B4%29%5E2
=>h=-4, k=-7, 4p=32=>p=8
vertex is at (h,k)=(-4,-7)->Q III
focus is at: (h,k%2Bp)=(-4,-7%2B8)=(-4,1)->Q II
=>focus and vertex lie in different quadrants
c.
x=-y%5E2%2F16-y%2F4%2B11%2F4
x=-%281%2F16%29%28y%5E2%2B4y%29%2B11%2F4....b=2
x=-%281%2F16%29%28y%5E2%2B4y%2B2%5E2%29-%28-%281%2F16%29%29%2A2%5E2%2B11%2F4
x=-%281%2F16%29%28y%2B2%29%5E2%2B1%2F4%2B11%2F4
x=-%281%2F16%29%28y%2B2%29%5E2%2B12%2F4
x=-%281%2F16%29%28y%2B2%29%5E2%2B3
x-3=-%281%2F16%29%28y%2B2%29%5E2+
-16%28x-3%29=%28y%2B2%29%5E2+
=>h=3, k=-2, 4p=-16=>p=-4
vertex is at (h,k)=(3,-2) ->Q VI
focus is at: (h%2Bp,k)=(3-4,-2)=(-1,-2) ->Q III
=>focus and vertex lie in different quadrants


d.
x=y%5E2%2F16%2By%2F4-19%2F4
x=%281%2F16%29%28y%5E2%2B4y%29-19%2F4
x=%281%2F16%29%28y%5E2%2B4y%2B2%5E2%29-%281%2F16%29%2A2%5E2-19%2F4
x=%281%2F16%29%28y%2B2%29%5E2-1%2F4-19%2F4
x=%281%2F16%29%28y%2B2%29%5E2-20%2F4
x=%281%2F16%29%28y%2B2%29%5E2-5
x%2B5=%281%2F16%29%28y%2B2%29%5E2
16%28x%2B5%29=%28y%2B2%29%5E2
=>h=-5, k=-2, 4p=16=>p=4
vertex is at (h,k)=(-5,-2) ->Q III
focus is at: (h%2Bp,k)=(-5%2B4,-2)=(-1,-2) ->Q III
=>focus and vertex lie in same quadrant


e.
x=-y%5E2%2F36-5y%2F18%2B299%2F36
x=-%281%2F36%29%28y%5E2%2B10y%29%2B299%2F36.........b=5
x=-%281%2F36%29%28y%5E2%2B10y%2B5%5E2%29-%28-%281%2F36%29%295%5E2%2B299%2F36
x=-%281%2F36%29%28y%2B5%29%5E2%2B25%2F36%2B299%2F36
x=-%281%2F36%29%28y%2B5%29%5E2%2B9
x-9=-%281%2F36%29%28y%2B5%29%5E2
-36%28x-9%29=%28y%2B5%29%5E2
=>h=9, k=-5, 4p=-36=>p=-9
vertex is at (h,k)=(9,-5) ->Q IV
focus is at: (h%2Bp,k)=(9-9,-5)=(0,-5) ->between Q III and Q IV
=>focus and vertex lie in different quadrants


f.
y=-x%5E2%2F24-5x%2F12%2B95%2F24
y=-%281%2F24%29%28x%5E2%2B10x%29%2B95%2F24
y=-%281%2F24%29%28x%5E2%2B10x%2B5%5E2%29-%28-1%2F24%29%2A5%5E2%2B95%2F24
y=-%281%2F24%29%28x%2B5%29%5E2%2B25%2F24%2B95%2F24
y=-%281%2F24%29%28x%2B5%29%5E2%2B5
y-5=-%281%2F24%29%28x%2B5%29%5E2+
-24%28y-5%29=%28x%2B5%29%5E2+

=>h=-5,+k=5, 4p=-24=>p=-6
vertex is at (h,k)=(-5,5)->Q II
focus is at: (h,k%2Bp)=(-5,5-6)=(-5,-1)->Q III
=>focus and vertex lie in different quadrants
parabola whose focus and vertex lie in different quadrant: b,c,e, and f


Answer by MathTherapy(10555) About Me  (Show Source):
You can put this solution on YOUR website!
Based on the equations of the parabolas, identify each parabola whose focus and vertex lie in different quadrants.
a. y=-x^2/8+x/4+23/8
b. y=x^2/32+x/4-13/2
c. x=-y^2/16-y/4+11/4
d. x=y^2/16+y/4-19/4
e. x=-y^2/36-5y/18+299/36
f. y=-x^2/24-5x/12+95/24
DEFINITELY not going to do all of those for you, JUST # 1.

If you like TORTURE, then you can do the problem the same way the other COMPLEX-PERSON who responded, did it! If you do not wish to TORTURE yourself, then read on.

matrix%281%2C3%2C+y%2C+%22=%22%2C+%28-+x%5E2%29%2F8+%2B+x%2F4+%2B+23%2F8%29

matrix%281%2C3%2C+8y%2C+%22=%22%2C+-+x%5E2+%2B+2x+%2B+23%29 ------- Multiplying by LCD, 8

matrix%281%2C3%2C+x%5E2+-+2x%2C+%22=%22%2C+-+8y+%2B+23%29

 ----- Complete the square on x by taking 1%2F2 of "b" on x, squaring it and ADDING it to both sides

 



Final equation, with a VERTICAL AXIS of SYMMETRY: highlight_green%28matrix%281%2C3%2C+%28x+-+1%29%5E2%2C+%22=%22%2C+-+8%28y+-+3%29%29%29 

                            Compare the above to: matrix%281%2C3%2C+%28x+-+h%29%5E2%2C+%22=%22%2C+4p%28y+-+k%29%29, where: