When looking at how functions behave, especially when we are checking their intercepts, asymptotes, and what happens at infinity, there are some simple steps that can help us understand better.

1. Find the Intercepts

First, let’s find the intercepts. These tell us where the graph crosses the axes.

• X-Intercepts: To find these, set the function to zero (like $f(x) = 0$) and solve for $x$.
• Y-Intercept: Plug in $0$ for $x$ (like $f(0)$) to find this intercept.

Example: For the function $f(x) = x^2 - 4$, we can find the x-intercepts by solving $x^2 - 4 = 0$, which gives us $x = -2$ and $x = 2$.

2. Look at Asymptotes

Next, we need to understand asymptotes. These help us see how the function behaves at the ends.

• Vertical Asymptotes: These are where the function is not defined. Check for any values that make the bottom part of a fraction zero (if working with fractions).
• Horizontal Asymptotes: These show what happens as $x$ gets really big or really small. Look at the biggest parts of the function.

Example: In the function $f(x) = \frac{1}{x-3}$, there’s a vertical asymptote at $x = 3$.

3. Look at Behavior at Infinity

Now, let's check what happens to the function when $x$ gets very large or very small.

• If the degree (the highest power) of the top part (numerator) is bigger than the bottom part (denominator), the function usually goes to infinity.
• If the bottom part has a higher degree, the function generally goes to zero.

4. Draw the Graph

With all this information, you can sketch the graph!

Plot the intercepts, draw the asymptotes, and show what happens at infinity. Connecting these points will help you see the shape of the function more clearly.

By using these easy steps, you can make it simpler to analyze any function’s behavior. This will help you draw better conclusions and create more accurate graphs!