To draw graphs with asymptotes correctly, you need to know what types of asymptotes there are. There are three main kinds:
Vertical Asymptotes: These happen where the function cannot be defined. For example, in the function ( f(x) = \frac{1}{x-2} ), there is a vertical asymptote at ( x = 2 ).
Horizontal Asymptotes: These show what happens to the graph when ( x ) gets really big. For the same function, as ( x ) gets bigger and bigger, ( f(x) ) gets closer to 0. So, there is a horizontal asymptote at ( y = 0 ).
Oblique Asymptotes: These come up in some functions when the top part (the numerator) has a higher degree than the bottom part (the denominator) by one. For example, in ( f(x) = \frac{x^2 + 1}{x} ), as ( x ) gets really big, the graph behaves like ( y = x ).
After finding the asymptotes, you should draw them on your graph. Then, check how the function behaves when you get close to those lines. Finally, connect the points smoothly, making sure to follow the rules of the asymptotes. This way, you will be able to sketch the functions correctly!
To draw graphs with asymptotes correctly, you need to know what types of asymptotes there are. There are three main kinds:
Vertical Asymptotes: These happen where the function cannot be defined. For example, in the function ( f(x) = \frac{1}{x-2} ), there is a vertical asymptote at ( x = 2 ).
Horizontal Asymptotes: These show what happens to the graph when ( x ) gets really big. For the same function, as ( x ) gets bigger and bigger, ( f(x) ) gets closer to 0. So, there is a horizontal asymptote at ( y = 0 ).
Oblique Asymptotes: These come up in some functions when the top part (the numerator) has a higher degree than the bottom part (the denominator) by one. For example, in ( f(x) = \frac{x^2 + 1}{x} ), as ( x ) gets really big, the graph behaves like ( y = x ).
After finding the asymptotes, you should draw them on your graph. Then, check how the function behaves when you get close to those lines. Finally, connect the points smoothly, making sure to follow the rules of the asymptotes. This way, you will be able to sketch the functions correctly!