Question

Please Help Constructing a Function for a parabola
See Attachment

Answer 1

well, let's us recall the parabola equation of \(\bf y=a(x-h)^2+k\)
those 2 parabolas are moving over the y-axis, vertically, thus the "x" will be the squared component

for g(x), the "a" component is positive, going upwards
for h(x), the "a" component is negative, going downwards

Answer 2

1 and -1 ?

Answer 3

well... that just means that \(\bf g(x) = a(x-h)^2+k \ and\ \ h(x) = -a(x-h)^2+k\)

Answer 4

ohh ok

Answer 5

the vertex is the parabola is at the coordinates (h, k) as you can see in the graph, both parabolas have their vertex in the origin, thus is (0, 0) or h = 0, k = 0 for both, thus \(\bf g(x) = a(x-h)^2+k\implies g(x) = a(x-0)^2+0\implies g(x)=ax^2\\ \quad \\
h(x) = -a(x-h)^2+k\implies h(x)=-a(x-0)^2+0\implies h(x)=-ax^2\)

Answer 6

so you'd ask.... what's "a"? well to get "a" we just need another point in the parabola.... taking a peek at the picture, look at the very corners..... g(x) =34 x = 4

so we can use that and say \(\bf g(x) = ax^2\implies 34=a(4)^2\implies ? = a\)

Answer 7

and we do the same for h(x), look at the bottom corners, when h(x) = -8, x = 4, thus \(\bf h(x) = -ax^2\implies -8=a(4)^2\implies ? = a\)

Answer 8

well \(\bf h(x) = -ax^2\implies -8=-a(4)^2\implies ? = a\)

Answer 9

would a be 0.5?

Answer 10

which one? for g(x)?

Answer 11

for h(x)

Answer 12

yes

Answer 13

or 1/2

Answer 14

so now we know that \(\bf h(x) = -ax^2\implies -8=a(4)^2\implies \cfrac{1}{2} = a\\ \quad \\
h(x) = -ax^2\implies h(x) = -\cfrac{1}{2}x^2\)

Answer 15

okay thanks and g(x) would be 17/8?

Answer 16

hmm ... wel... is not 34 for one, so much for my typo.... is 32... so \(\bf g(x) = ax^2\implies 32=a(4)^2\implies \cfrac{32}{16} = a\implies 2=a\\ \quad \\
g(x) = ax^2\implies g(x) = 2x^2\)

Answer 17

http://fooplot.com/#W3sidHlwZSI6MCwiZXEiOiIyeF4yIiwiY29sb3IiOiIjRDkxRkFFIn0seyJ0eXBlIjowLCJlcSI6IigtMS8yKXheMiIsImNvbG9yIjoiIzE4NDNGMiJ9LHsidHlwZSI6MTAwMH1d =)

Answer 18

I think its just a computer error

Answer 19

hmm am I wrong or you have their order backwards? you put the g(x) on the h(x) slot and the other on the other

Answer 20

you were right. I made a mistake. Sorry about that. Thank you for walking me through it and for the link

Answer 21

yw