double parabolas

Question

Functions f and g are represented by parabolas.

Function f:

1) f(0) = 150

2) function f is increasing over the interval [-16, infinity [

3) the minimum of function f is -10

Function g:

1) Positive over the interval [x, -14 ]

2) g(-24) = -500

3)Functions f and g have one zero in common where f(x)=g(x)=0

I dont really know where to start...

Answer 1

double parabola is double trouble

f has minimum = -10 = vertex

y = a(x-h)^2 -10 in vertex form

y(0) = f(0) = 150 = a(x-h)^2 -10

a(x-h)^2 = 160

(x-h)^2 = 160/a

x-h = +/-4sqr(10/a)

x = h +/-4sqr(10/a)

increasing over interval x>-16

x = -16+/-4sqr(10/a)

let a =5/8

then

x = 0 then

f(0) = (5/8)(0+16)^2 -10 5(16)(16)/8 -10 = 150

f(x) = (5/8)(x+16)^2 -10

x is a zero when f(x) = 0 = (5/8)(x+16)^2 -10

(8/5)(10) = (x+16)^2

x+16 =-16 +/-sqr16 =-16 +/-4

x = -12 or -20

do similar manipulations for g(x), then set what you get equal to f(x) =0, solve for x, you should get either -12 or -20, or both or no solution. graphing the points suggests No solution that fits the requirements Unless g(x) is a downward opening vertical parabola with a<0

g(x) = y(x) =a(x-h)^2 +k

g(-24) = -500, the point (-24,-500)

g(0) = -14, as y>0 for all x>-14 and y=0 for x=-14, the point (0, -14)