Sketch the curve f(x)=2x^2-3x/x^2-x-12 by providing a full analysis of its features: (10 marks)
x-intercept(s):
y-intercept(s):
Vertical Asymptote(s):
Horizontal Asymptote(s):
End behaviour near Horizontal Asymptotes:
As x --> , y -->_______
As x --> - , y -->_______
End behaviour near Vertical Asymptotes:
(Show all your work)
To analyze the curve of the function f(x) = (2x^2 - 3x) / (x^2 - x - 12), we will need to find the x-intercepts, y-intercepts, vertical asymptotes, horizontal asymptotes, and determine the end behavior near the asymptotes.
1. X-Intercepts:
The x-intercepts occur when the value of y is equal to zero. To find them, set f(x) = 0 and solve for x:
(2x^2 - 3x) / (x^2 - x - 12) = 0
The numerator can be factored as: x(2x - 3), and the denominator can be factored as: (x - 4)(x + 3).
So we have:
x(2x - 3) / (x - 4)(x + 3) = 0
This equation will be true if either the numerator, x(2x - 3), is equal to zero, or if the denominator, (x - 4)(x + 3), is equal to zero.
Set numerator equal to zero:
x(2x - 3) = 0
Solving for x, we get:
x = 0 (x-intercept)
Set denominator equal to zero:
(x - 4)(x + 3) = 0
Solving for x, we get:
x = 4 (x-intercept)
x = -3 (x-intercept)
So the x-intercepts are x = 0, x = 4, and x = -3.
2. Y-Intercept:
The y-intercept is the point where the curve intersects the y-axis, which occurs when x is equal to zero.
To find it, substitute x = 0 into the equation f(x):
f(0) = (2*0^2 - 3*0) / (0^2 - 0 - 12)
f(0) = 0 / (-12)
f(0) = 0
So the y-intercept is y = 0.
3. Vertical Asymptotes:
Vertical asymptotes occur where the denominator is equal to zero. To find them, set the denominator equal to zero:
x^2 - x - 12 = 0
This equation can be factored as (x - 4)(x + 3) = 0
So the vertical asymptotes are x = 4 and x = -3.
4. Horizontal Asymptotes:
To find the horizontal asymptotes, we need to evaluate the limit of f(x) as x approaches positive infinity and negative infinity.
As x approaches positive infinity, we divide the leading terms of the numerator and denominator by the highest power of x (2x^2):
lim(x->∞) (2x^2 - 3x) / (x^2 - x - 12) = 2
So there is a horizontal asymptote at y = 2.
As x approaches negative infinity, we do the same:
lim(x->-∞) (2x^2 - 3x) / (x^2 - x - 12) = 2
So there is also a horizontal asymptote at y = 2.
5. End Behavior:
As x approaches positive infinity, the end behavior of f(x) is y approaches 2.
As x approaches negative infinity, the end behavior of f(x) is y approaches 2.
In summary:
- X-intercepts: x = 0, x = 4, x = -3
- Y-intercept: y = 0
- Vertical Asymptotes: x = 4, x = -3
- Horizontal Asymptote: y = 2
- End behavior: As x approaches positive/negative infinity, y approaches 2.