f:x -> (4x²-8x-21)/(x²-1)
(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
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.
To analyze the given function f(x) = (4x²-8x-21)/(x²-1), you need to consider the domain, asymptotes, zeros, and extrema. Let's break it down step by step:
(a) Domain:
The domain of a function is the set of all possible input values (x-values) for which the function is defined. In this case, the only restriction on the domain is that x cannot be equal to 1. Therefore, the domain of f(x) is all real numbers except x=1.
(b) Asymptotes:
To find the asymptotes, you need to investigate what happens as x approaches certain values. In this case, we have a rational function with a quadratic numerator and denominator.
Vertical Asymptote:
Vertical asymptotes occur where the denominator of a rational function equals zero (x²-1 = 0). So, solving this equation, we get x=1 and x=-1. However, since x=1 is not in the domain, it is not a vertical asymptote for f(x). Therefore, the only vertical asymptote for f(x) is x=-1.
Horizontal Asymptote:
To determine the horizontal asymptote, compare the degrees of the numerator and denominator. In this case, both the numerator and denominator have a degree of 2. To find the horizontal asymptote, divide the leading coefficients of the numerator and denominator. Here, it is 4/1 = 4. Therefore, the horizontal asymptote for f(x) is y=4.
(c) Zeros:
Zeros of a function are the values of x for which the function equals zero. To find the zeros of f(x), set the numerator equal to zero:
0 = 4x²-8x-21
This is a quadratic equation, which you can solve either by factoring or using the quadratic formula. If factoring is not apparent, you can use the quadratic formula:
x = (-b ± √(b²-4ac))/2a
For this equation, a=4, b=-8, and c=-21. Plugging into the quadratic formula, you can find the values of x that make f(x) equal to zero.
(d) Extrema:
To find the extrema (minimum or maximum points) of f(x), you need to differentiate the equation with respect to x and then set the derivative equal to zero. However, in this case, it might be challenging to differentiate the function in its current form. One way to simplify it is by decomposing it into partial fractions.
To sketch the graph, start by plotting the vertical asymptote at x=-1. Then, consider the zeros you found earlier and plot those points on the graph. Also, remember that you have the horizontal asymptote y=4. Finally, take note of the behavior of the function as x approaches positive or negative infinity to complete your sketch.
I hope this guidance helps you analyze the given function and sketch its graph!