Analyzing how a function behaves before drawing its graph is really important for a few reasons:

1. Finding Important Parts:

   • Intercepts: To find the y-intercept, calculate ( f(0) ).
   • Roots: To find the x-intercepts, solve ( f(x) = 0 ).

2. Behavior at Infinity:

   • Look at what happens to the function when ( x ) gets really big (positive infinity) or really small (negative infinity). This helps us understand how the graph behaves at the ends.

3. Critical Points and Highs and Lows:

   • Use the first derivative ( f'(x) ) to spot the high points (maxima) and low points (minima).
   • Use the second derivative ( f''(x) ) to figure out how the curve bends (concavity).

4. Breaks in the Function:

   • Find any spots where the function isn't defined, like where there are vertical lines (asymptotes).

Doing this analysis helps create a clear and helpful graph, making it easier to understand how the function works overall.