Interpreting Sign Charts
This page follows the JoeCMath video What You NEED to Know About Sign Charts. The video focuses on what a sign chart tells you after it has already been made, especially when you compare the sign chart for a polynomial with sign charts for its first and second derivatives.
Main Idea
A sign chart is useful because the signs have meaning. The important part is knowing what each sign chart is describing.
| Sign chart | Positive means | Negative means |
|---|---|---|
| $f(x)$ | The graph is above the $x$-axis. | The graph is below the $x$-axis. |
| $f^{\prime}(x)$ | The graph is increasing. | The graph is decreasing. |
| $f^{\prime\prime}(x)$ | The graph is concave up. | The graph is concave down. |
The sign chart for the function tells you where the graph lives relative to the $x$-axis. The first derivative sign chart describes motion, not height. The second derivative sign chart describes bending, not whether the graph is high or low.
Interpreting The Function Sign Chart
For the original polynomial $f(x)$, the signs tell you whether the graph is above or below the $x$-axis:
\[f(x)>0 \quad \Rightarrow \quad \text{above the } x\text{-axis}\]and
\[f(x)<0 \quad \Rightarrow \quad \text{below the } x\text{-axis}.\]A zero of $f(x)$ is an $x$-intercept. The sign chart tells you what happens between those intercepts.
Watch this part: interpreting the polynomial function sign chart at 0:25.
Interpreting The First Derivative Sign Chart
For $f^{\prime}(x)$, the signs tell you whether the graph is increasing or decreasing:
\[f^{\prime}(x)>0 \quad \Rightarrow \quad f(x) \text{ is increasing}\]and
\[f^{\prime}(x)<0 \quad \Rightarrow \quad f(x) \text{ is decreasing}.\]A sign change in $f^{\prime}(x)$ can mark a turning point:
| Sign change in $f^{\prime}(x)$ | Graph behavior |
|---|---|
| $+$ to $-$ | Local maximum candidate |
| $-$ to $+$ | Local minimum candidate |
Watch this part: interpreting the first derivative sign chart at 1:54.
Interpreting The Second Derivative Sign Chart
For $f^{\prime\prime}(x)$, the signs tell you the concavity:
\[f^{\prime\prime}(x)>0 \quad \Rightarrow \quad f(x) \text{ is concave up}\]and
\[f^{\prime\prime}(x)<0 \quad \Rightarrow \quad f(x) \text{ is concave down}.\]A sign change in $f^{\prime\prime}(x)$ can mark an inflection point, where the graph changes how it bends.
Watch this part: interpreting the second derivative sign chart at 3:28.
Four Shape Combinations
The video also organizes the first and second derivative signs together. Each interval can be described by both its motion and its bending.
| $f^{\prime}(x)$ | $f^{\prime\prime}(x)$ | What the graph does |
|---|---|---|
| $+$ | $+$ | Increasing and concave up |
| $+$ | $-$ | Increasing and concave down |
| $-$ | $+$ | Decreasing and concave up |
| $-$ | $-$ | Decreasing and concave down |
This is the bridge from sign charts to graph shape. You are not just plotting points; you are building a template for how the curve should move through each interval.
Using The Charts Together
When all three sign charts are available, read them in layers:
- Use $f(x)$ to decide whether the graph is above or below the $x$-axis.
- Use $f^{\prime}(x)$ to decide whether the graph is increasing or decreasing.
- Use $f^{\prime\prime}(x)$ to decide whether the graph is concave up or concave down.
- Combine those clues with the actual zeros and intercepts to sketch the polynomial.
The video ends by using the sign chart template together with the graph of the polynomial, so the chart is not separate from graphing. It becomes a way to organize what the graph is supposed to do.
Watch this part: using the sign chart template with the graph at 7:12.
Timestamp Guide
Related Sign Chart Work
If you want the construction process before the interpretation step, use the companion guide for polynomial sign charts.
Related JoeCMath clips: