f:x -> (4x²-8x-21)/(x²-1)

Question

(a) Analyse
(b) Sketch

I'm struggling on (a) so I can't do (b)
Here are my ideas/what I've got so far:

(a)Domain, x is a set of real numbers; x can not equal 1

Asymptotes x=1, y=?
I think you have to divide here but i can't figure it out with the x² on the bottom.

zeroes: when x=0, y=21
when y=0, x=?
0=4x²-8X-21 then??

extrema; differentiate eqn here...not sure how (maybe convert to another form?)

I think that is all for the analysis?

For any help, thanks in advance

Answer 1

where to begin ? ....
f:x -> (4x²-8x-21)/(x²-1) corresponds to the equation
y = (4x²-8x-21)/(x²-1)

for vertical asymptotes, set the denominator equal to zero
so x^2 - 1 = 0
(x+1)(x-1) = 0
x = ± 1
so there are two vertical asymptotes
at x = 1 and x = -1

for horizontal asymptotes, let x ---> ∞

for very large values of x, the signigicant part of the equation is
y = 4x^2 / x^2 which approaches 4
so y = 4 is the horizontal axis

for x-intercepts, let y = 0
so (4x²-8x-21)/(x²-1) = 0
(4x²-8x-21)= 0
(2x+3)(2x-7) = 0
x = -3/2 and x = 7/2 are the x-intercepts

for y -intercepts , let x = 0 in the equation
so y = (0-0-21)/(0-1) = 21
so the y-intercept is 21

Derivative: by quotient rule
dy/dx = [(x^2-1)(8x-8) - 2x(4x^2-8x-21)]/(x^2 - 1)^2
expanding the top and setting this equal to zero gave me

8x^2 + 34x + 8 = 0

solve this using the quadratic formula to get the x's of the extrema,
sub those x's back in the origianl to get their y's.