Curve Sketching
Now we put it all together. Given a formula, you can draw an accurate picture of its graph
without plotting hundreds of points — just a handful of carefully chosen facts. Each
tool you've met fills in one feature:
- Intercepts: where the curve meets the axes (y-intercept at x=0; x-intercepts where f(x)=0).
- f' sign: where it rises and falls.
- Extrema: the peaks and valleys.
- Concavity & inflection: how it bends, and where the bend flips.
A worked sketch
Let's sketch f(x) = x^3 - 3x from scratch.
\begin{aligned}
&\textbf{Intercepts:}\ f(0)=0;\quad x^3-3x = x(x^2-3)=0 \Rightarrow x = 0,\ \pm\sqrt3 \\
&\textbf{Slope:}\ f'(x)=3x^2-3=0 \Rightarrow x=\pm1,\quad \text{max at } (-1,2),\ \text{min at } (1,-2) \\
&\textbf{Bend:}\ f''(x)=6x=0 \Rightarrow \text{inflection at } (0,0)
\end{aligned}
Plot those few points, mark the curve rising–falling–rising, and bend it down then up. The
sketch practically draws itself. Step through it below.
The three graphs at once
It's worth seeing how f (bold), f' and
f'' line up. Where f'=0 the bold curve
has a peak or valley; where f''=0 it has an inflection point.
Slide across and watch the three readings agree.
Watch it on Khan Academy