Question

Help please :)
Find the values of a and b if the parabola y=a(x+b)^2 −8 has tangent y = 2x at the point P(4, 8),

Answer 1

i have gotten gradient to be 2a(x+b)

Answer 2

ok great so far

y' at (x,y)=(4,8) is 2a(4+b)=8a+2b
And we are also given at that point we have y'=2x
But that point is (4,8) so at that point we have y'=8
So this means we have 8=8a+2b

Did you get this far?

Answer 3

We should be looking for another liner equation

Answer 4

Any ideas?

Answer 5

not really

Answer 6

i got 8=8a +2ab

Answer 7

Well you also know (4,8) is on the parabola. See if you can use that.

Answer 8

this should give you two equations with the only unknowns a and b

Answer 9

You know how to solve a system of equations, correct?

Answer 10

err yea

Answer 11

Did you think to do 8=a(4+b)^2-8

Answer 12

i got 8=8a +2ab not 8=8a +2b btw

Answer 13

wait how did you get the first equation?

Answer 14

oh wait i messed up when i wrote it

Answer 15

2a(4+b)=8a+2b

I didn't distribute there correctly

8a+2ab would be correct

Answer 16

so yeah we have 8a+2ab=8

Answer 17

2nd equation is 16a + 8ba + ab^2 - 16 right?

Answer 18

all of that = 0 yes

Answer 19

then we just solve right?

Answer 20

yes

Answer 21

16a + 8ba + ab^2 - 16 = 0 and 8a+2ab-8= 0
err how?

Answer 22

Did you try solving the linear equation either for a or b
and replace the a or b in the first equation you ahve there

Answer 23

im pretty sure we made a mistake somewhere

Answer 24

http://www.wolframalpha.com/input/?i=16a+%2B+8ba+%2B+ab%5E2+-+16+%3D+0++and++8a-2ab-8%3D+0

Answer 25

the answers in the back of my book says 1/16 = a and 12= b

Answer 26

why did you change the sign in that one equation?

Answer 27

oh typed it wrong before

Answer 28

oh wait...

Answer 29

can we start from the begining again?

Answer 30

the derivative is 8a+2ab
that means the gradient is 8a + 2ab
How did it become
8= 8a + 2ab

Answer 31

y'=2x at (4,8)

Answer 32

we also know that y'=2a(x+b) at (4,8)

so at (4,8) we know 2x=2a(x+b)

Answer 33

x=4 then

Answer 34

so we have 2*4=2a(4+b)

Answer 35

8=8a+2ab

Answer 36

ok alright

Answer 37

your book's answer doesn't make sense

Answer 38

hmm

Answer 39

so if a=1/16, b=12
and we have that y'=2x at (4,8) and y'=2a(x+b) at (4,8)
then we have

2*4=2*1/16(4+12)
correct?
is this true?

Answer 40

2*4=8
2/16*(4+12)=1/8*(16)=2
2 does not equal 8

Answer 41

err yea

Answer 42

The equation you put is correct right?
y=a(x+b)^2-8

Answer 43

the point is (4,8) where we have the tangent line y=2x

Answer 44

yes right question

Answer 45

i see nothing wrong with our resulting equations

Answer 46

hmm yea

Answer 47

but i dont think we could sub (4,8) into 16a + 8ba + ab^2 - 16=0

Answer 48

we already did to get that equation

Answer 49

remember y=a(x+b)^2-8
(4,8) is on that parabola

Answer 50

8=a(4+b)^2-8

Answer 51

a(4+b)^2-16=0
a(16+8b+b^2)-16=0
ab^2+8ab+16a-16=0

Answer 52

So we have


ab^2+8ab+16a-16=0
4=4a+ab

Answer 53

Solve that bottom equation for either a or b then plug into the first equation.

Answer 54

a = 1, b=0

Answer 55

By the way I just divided our earlier linear equation by 2 to get 4=4a+ab

4=a(4+b)
4/(4+b)=a

Assuming b does not equal -4.

Then we have 0=4/(4+b)*b^2+8*(4/(4+b))*b+16*(4/(4+b))-16

Answer 56

Multiply both sides by 4+b

0=4b^2+32b+64-16(4+b)

Answer 57

http://au.answers.yahoo.com/question/index?qid=20130720230137AA4cCvd

Answer 58

yep so i did make a mistake there
the slope of 2x is 2
I put 8. haha
So we had 2=8a+2ab was suppose to be that one linear equation
not 8=8a+2ab