Step 1: Find $\textcolor[rgb]{}{f}\textcolor[rgb]{}{\prime}\textcolor[rgb]{}{\left(}\textcolor[rgb]{}{x}\textcolor[rgb]{}{\right)}$and $\textcolor[rgb]{}{f}\textcolor[rgb]{}{\prime}\textcolor[rgb]{}{\prime}\textcolor[rgb]{}{\left(}\textcolor[rgb]{}{x}\textcolor[rgb]{}{\right)}$.
Step 2: Run the First 3-Steps of the 1 ^{st} Derivative Test for Intervals of Increasing and Decreasing.
Step 3: Run the First 3-Steps of the 2 ^{nd} Derivative Test for Intervals of Concave Up and Concave Down.
Step 4: Create a combined number line bringing together the data from your 1 ^{st} Derivative Test number line and your 2 ^{nd} Derivative Test number line.
Step 5: Read the shape of the curve based upon your combined number line, and draw it in the $\textcolor[rgb]{}{f}\textcolor[rgb]{}{\left(}\textcolor[rgb]{}{x}\textcolor[rgb]{}{\right)}$section of your chart.
There are two sides to each concavity shape , an increasing side and a decreasing side. Use the 1 ^{st} Derivative, $\textcolor[rgb]{}{f}\textcolor[rgb]{}{\prime}\textcolor[rgb]{}{\left(}\textcolor[rgb]{}{x}\textcolor[rgb]{}{\right)}$, data to determine if the graph is + = increasing or – = decreasing, and then use the 2 ^{nd} Derivative, $\textcolor[rgb]{}{f}\textcolor[rgb]{}{\prime}\textcolor[rgb]{}{\prime}\textcolor[rgb]{}{\left(}\textcolor[rgb]{}{x}\textcolor[rgb]{}{\right)}$, data to know if it is + = concave up or – = concave down.
Step 6: Sketch the curve on your graph taking the shapes you drew on your number line, and connecting them at the critical points and inflection points .
Often times you will need to plug the x-values of your critical values and inflection points back into the original equation, $\textcolor[rgb]{}{f}\textcolor[rgb]{}{\left(}\textcolor[rgb]{}{x}\textcolor[rgb]{}{\right)}$, to get the y-values that go with those x-values .
The critical points and inflection points are where your graph is changing behavior.