Interpreting Sign Charts
Simple overview on how to interpret a polynomial sign chart.
Main Idea
For any given polynomial $f(x)$, the sign chart of the function $f(x)$, the first derivative $f^{\prime}(x)$, and the second derivative $f^{\prime\prime}(x)$ tell us a lot about the shape of the polynomial when we go to sketch it free hand like the cavemen did.
| Sign chart | Positive Interval means | Negative Interval 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 ($f(x)$) tells us when the function lives above or below the $x$-axis.
The first derivative ($f^{\prime}(x)$) sign chart tells us when the function is increasing or decreasing (think the slope at any given point).
The second derivative ($f^{\prime\prime}(x)$) sign chart describes the concavity of the curve in the interval (concave up like $\cup$, while concave down looks like $\cap$).
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.
Please watch the video to see them all used together to get an idea of how these all work together. I’ll probably revisit this page to detail it back out better in the future, but today I sort of feel 🤷.
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: