Is it possible to roughly sketch a graph(without plotting the points) from the asymptotes of a rational function? Is yes, what are the rules?
For example, if I have to sketch a graph of y = (2x^2+10x-12)/x^2+x-6), how do I do it?
I got the vertical asymptotes as x=-3 & x=2, and horizontal asymptote y=2.
How do I draw a graph of this function without plotting points?
Thanks
Factor the numerator and denominator.
find the roots of the numerator...that is where the graph will cross x axis.
Find the roots of the denomiantor, those are the points where y will be +- inf
sketch in the horizontal asy...
Now SKETCH in the missing areas.
To sketch a graph of a rational function using the asymptotes, follow these steps:
1. Factor the numerator and denominator:
In your example, the numerator is 2x^2 + 10x - 12 and the denominator is x^2 + x - 6. Factor them as much as possible:
Numerator: 2x^2 + 10x - 12 = (2x - 2)(x + 6)
Denominator: x^2 + x - 6 = (x + 3)(x - 2)
2. Find the roots of the numerator:
Set the numerator equal to zero and solve for x to find the x-intercepts (where the graph will cross the x-axis):
(2x - 2)(x + 6) = 0
2x - 2 = 0 or x + 6 = 0
x = 1 or x = -6
3. Find the roots of the denominator:
The roots of the denominator represent the vertical asymptotes of the function. Set the denominator equal to zero and solve for x:
(x + 3)(x - 2) = 0
x + 3 = 0 or x - 2 = 0
x = -3 or x = 2
So, the vertical asymptotes are x = -3 and x = 2.
4. Determine the horizontal asymptote:
The horizontal asymptote can be found by comparing the degrees of the numerator and denominator. Since both have the same degree (2), divide the coefficient of the leading term in the numerator (2) by the coefficient of the leading term in the denominator (1). The horizontal asymptote is the quotient obtained, which in this case is y = 2.
5. Sketch the horizontal asymptote:
Draw a horizontal line at y = 2 on your graph, as this represents the horizontal asymptote.
6. Sketch in the missing areas:
Now that you know the x-intercepts and vertical asymptotes, you can sketch the general shape of the graph. Connect the points smoothly and make sure the graph approaches the vertical asymptotes as they get closer to them.
By following these steps, you should be able to rough sketch the graph of the rational function without plotting individual points.