Theory
Note! Whatever kind of function you have, the sign charts tell you where the function you’re drawing sign charts for is above or below the $x$-axis.
The main reason to love sign charts is that the sign chart of a differentiated function tells you how the original function behaves.
Rule
This rule applies for drawing any sign chart.
You draw a solid line when the $y$-value of the function is greater than zero, which is when $f(x)>0$.
You draw a dashed line when the $y$-value of the function is less than zero, which is when $f(x)<0$.
Example 1
In the figure below, you can see that the graph is below zero up to $x=2$ and between $x=5$ and $x=8$. In these intervals the sign chart is dashed. The graph is above the $x$-axis between $x=2$ and $x=5$ and when $x>8$. In these areas the sign chart is solid.
Note! When you draw sign charts for constants, you just draw a solid line for positive numbers and a dashed line for negative numbers.
But how do you know where the function is above or below the $x$-axis? Here are two ways to find out. Use Method 1 when you have a linear expression. In other cases you can use the one you like best.
Rule
Example 2
Draw the sign chart of $f(x)=-2x+12$
Rule
Example 3
Find the zeros of the function
$$f(x)={x}^{2}-7x+12$$ |
and decide where the graph is above or below the $x$-axis
$$(x-4)(x-3).$$ |
First, you find where $x-3=0$: $$\begin{array}{llll}\hfill x-3& =0\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill x& =3\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\end{array}$$
That means you write 3 on the number line on top of the sign charts.
Then you find where the sign chart is positive and negative:
Choose a value smaller than 3, for example $x=0$, and insert it into $x-3$. Then you get $0-3=-3<0$, which means the line to the left of 3 should be dashed.
Choose a value greater than 3, for example $x=10$, and insert it into $x-3$. Then you get $10-3=7>0$, which means the line to the right of 3 should be solid.
Draw this sign chart below the number line.
First, you find where $x-4=0$: $$\begin{array}{llll}\hfill x-4& =0\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill x& =4\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\end{array}$$
That means you write 4 on the number line on top of the sign charts.
Then you find where the sign chart is positive and negative:
Choose a value smaller than 4, for example $x=0$, and insert it into $x-4$. Then you get $0-4=-4<0$, which means the line to the left of 4 should be dashed.
Choose a value greater than 4, for example $x=10$, and insert it into $x-4$. Then you get $10-4=6>0$, which means the line to the right of 4 should be solid.
Draw this sign chart below the sign chart for $(x-3)$.
The graph is above the $x$-axis on the interval
$$\phantom{\rule{-0.17em}{0ex}}\left(-\infty ,3\right)\cup \phantom{\rule{-0.17em}{0ex}}\left(4,\infty \right),$$ |
and it’s below the $x$-axis on the interval $\phantom{\rule{-0.17em}{0ex}}\left(3,4\right)$. When $x=3$ and $x=4$, the graph is neither above or below the $x$-axis.