When you want to draw a graph of a function, there are some important points to look out for. These points help you make a clear and accurate picture:

• Intercepts:

  • The $y$-intercept is the point where the graph crosses the $y$-axis. You find it by putting $x = 0$ into the function.
  • The $x$-intercepts are where the graph hits the $x$-axis. You find these points by setting the function equal to zero, or $f(x) = 0$.

• Critical Points:

  • These points happen where the first derivative, which tells us about the slope, is either zero or doesn’t exist.
  • Critical points are important for figuring out where the graph reaches its highest or lowest points.

• Behavior as $x$ gets really big or really small:

  • Looking at how the function acts when $x$ gets large (going to positive infinity) or very small (going to negative infinity) helps to understand how the graph looks far left and right.
  • For polynomial functions, this usually connects to the leading coefficient and the degree of the polynomial.

• Asymptotes:

  • Vertical asymptotes are lines where the function doesn't exist, usually where the denominator is zero.
  • Horizontal asymptotes show how the function behaves at the ends and are often found by comparing the highest powers in the numerator and denominator.

• Increasing and Decreasing:

  • By checking what happens around critical points, you can find out where the function is going up or down. This helps you see the shape of the graph.

• Concavity and Inflection Points:

  • To find inflection points, where the graph's curve changes, use the second derivative.
  • This information helps you make the graph even more accurate.

• Key Values:

  • Evaluating the function at important values, like $x = -1, 1, 2$, gives you points that can help anchor your graph.

By carefully looking for these key points, you can create a clear and detailed sketch of the function.