Question

can someone help me!!??? please i would really appreciate it. question: If y=-4x²+kx-1determine the value(s) of k for which the maximum value of the function is an integer. Explain your reasoning using pictures, numbers, and words.

Answer 1

lets try this from the quad formula perspective...

sqrt(k^2 -4(-1)(-4)) would have to be equal to or greater than 0

Answer 2

k^2 -16 >= 0

k^2 >= 16

k >= +-sqrt(16)

k>= -4, or 4

Answer 3

or rather k>= |4 ro -4|...might be a better scenario :)

Answer 4

since -4x^2 gives us an upsode down "U" for a graph, we can determine the "largest value"... at the vertex:

-k/2(-4)

Answer 5

-k/-8 = k/8

we have some value for "x" which is equal to k/8

can we use that in our equation?

Answer 6

y=-4(k/8)²+k(k/8)-1

y = -4k^2/61 + k^2/8 - 1

Answer 7

61 = 64 in my world lol

Answer 8

y = -4k^2/64 + 8k^2/64 -1

y = 4k^2/64 - 1

y = 4k^2/64 - 1

y = k^2/16 - 1

can we do anything with this "value"?

Answer 9

or am I making this to hard on myself....

Answer 10

-4x^2 +kx -1 = 0 factors to..

4x^2 -kx +1 = 0

1,4 = 4
2,2 = 4

(x- 1)(4x- 1)

or

(2x-1) (2x-1)

Answer 11

k=5.... i beieve

Answer 12

how??

Answer 13

the values we get for the product of "ac" are:

4(1) = 4

1+4 = 5, k=5 in this instance

2+2 = 4, k=4 in this instance.

but I might have a "sign" out of place so I would have to recheck it all :)

but does that make sense?

Answer 14

can we do this from the beginning again??

Answer 15

if y=-4x^2 +k -1 dont we plug it into the discriminant?

Answer 16

a=-4 b=k c=-1

Answer 17

which beginning :) I was trying to determine a good manner in which to appraoch the problem...the discriminate there was not an "easy" way.... that I could see.

Answer 18

i mean from the beggining of the question...so far thats what i know

Answer 19

but the question states tht the value of the function is an integer so does tht mean that the value of k <0

Answer 20

its easier to approach from the "meaing" that is given to a,b, and c

the "c" term is a product of 2 numbers;

the "b" term is a sum of 2 numbers.

for "b" to have the greatest value, the sum of the factors of "c" would have to combine to the biggest value right?

Answer 21

i have no idea what u just said?? lol

Answer 22

an integer is any number between -infinity and +infinity... :)

Answer 23

ooo okay...

Answer 24

the bigest integer value can p[ossibly be a negative value as long as it is the biggest integer we can get :)

Answer 25

but lets try this approach...

first we factor out that -1 to make our lives easier:

y = -1(4x^2 -kx +1) we good here?

Answer 26

okay..

Answer 27

we need to gather a "pool" of options for the middle term. that "pool" comes from the number we get when we multiply "a" and "c".

4 times 1 = 4. we good so far?

Answer 28

wait when u r doing this r u plugging it into the discriminant??? or just doing it...?

Answer 29

just doing it.... the discrimant is only valueable if you "know" what k is.... so lets continue on this path and see where it leads :)

Answer 30

we need two number: call them "m" and "n", that multiply together to get "4" and ADD together to get our value for "k". does that make sense?

Answer 31

yes ur doing product n sum

Answer 32

correct:

our options are:

4(1) = 4; 4+1 = 5

2(2) = 4; 2+2 = 4

which number is greater.... 4 or 5?

Answer 33

the 2(2) seems reasonable..cuz u need greates common factor right??

Answer 34

we couldnt care less about "common" factors here :)

what we care about is the greatest value that we can get for "k".

the value for "k" is obtained by "adding together" the factors that make up "4". That is our only concern :)

Answer 35

so if we r looking for the greatest common value then we use the numbers 1 n 4...

Answer 36

i mean the greatest value for k

Answer 37

thats right.... so lets see if 5 is a good answer by plugging it into our original formula and seeing what we get:)

y = -4x^2 +(5)x - 1 how would we TEST this situation?

Answer 38

if y =0??

Answer 39

our "highest point" is at our vertex: -b/2a

-5/2(-4) = -5/-8 = 5/8

5/8...our "vertex is at 5/8. what is the value of y at x = 5/8?

Answer 40

where did u get -b/2a?

Answer 41

that is from the quadratic formula as well.... do you know the quadratic formula?

Answer 42

yes

Answer 43

i didnt tht it gives u the highest point at the vertex..

Answer 44

i dint know

Answer 45

the quad formula says that the roots of a quadratic expression are equal distances from a certain value of x.... right?

-b sqrt(b^2-4ac)
--- + or - -------------
2a 2a

right?

Answer 46

yes

Answer 47

so the cernter, or vertex, of our equation lies between these 2 points..... the x value of our "vertex" then must be -b/2a.... am I right?

Answer 48

if you move 5 feet to your left, or 5 feet to your right.... where is the middle of that?.... exactly where you were standing to begin with right?

Answer 49

-sqrt(b^2-4ac).......-b.......+sqrt(b^2-4ac)
------------ --- -------------
2a 2a 2a

I hope this displays correctly lol

Answer 50

does that make sense?

Answer 51

yupp

Answer 52

i think i got it thanks i think i can handle it from here

Answer 53

thanks for ur help

Answer 54

ok :) but if this might clarify things a bit