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How Do Sign Charts Work?

Theory

What’s a Sign Chart?

A sign chart tells you when the value of a function $f (x)$ is negative or positive, which is the same as when the graph of $f (x)$ is below or above the $x$ -axis, respectively.

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

How to Draw Sign Charts

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.

A graph that moves above and below the x-axis with its corresponding sign chart.

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

Method 1: Deciding Whether the Function is Positive or Negative

1.: Solve the equation $f (x) = 0$ to find the points of intersections between the graph and the $x$ -axis.
2.: Look at one $x$ -value on each side of these intersections. If the $x$ -value gives you a negative $y$ -value, the graph is below the $x$ -axis in that area. If the $x$ -value gives you a positive $y$ -value, the graph is above the $x$ -axis in that area.
3.: Draw the sign chart.

Example 2

Draw the sign chart of $f (x) = - 2 x + 12$

1.: Solve the equation $- 2 x + 12 = 0$ : $\begin{array}{l} - 2 x + 12 & = 0 \\ - 2 x & = - 12 | \div (- 2) \\ x & = 6 \end{array}$
2.: Now you check one value of $x$ on each side of $x = 6$ , one that is greater than 6 and one that is smaller than 6. When you choose these numbers, it’s smart to pick numbers that are easy to deal with, like $x = - 10$ and $x = 10$ . $\begin{array}{l} f (- 10) & = - 2 \cdot (- 10) + 12 \\ = 20 + 12 \\ = 32 \\ > 0 \Rightarrow above the x -axis \\ f (10) & = - 2 \cdot (10) + 12 \\ = - 20 + 12 \\ = - 8 \\ < 0 \Rightarrow below the x -axis \end{array}$
3.: Draw the sign chart:

Rule

Method 2: Deciding Whether the Function is Positive or Negative

1.: Begin by solving the equation $f (x) = 0$ to find the intersections between the graph and the $x$ -axis.
2.: Factorize the expression.
3.: Draw a number line for the values of $x$ at the top of the sign charts.
4.: Draw a sign chart for each factor $(x - x_{i})$ , where $x_{i}$ are the values of $x$ at the points of intersections with the $x$ -axis.
5.: You know that $x - x_{i} = 0$ when $x = x_{i}$ , so you can mark $x_{i}$ on the number line on top, and put 0 under $x_{i}$ in the sign chart for $x - x_{i}$ .
6.: To find out where the function is positive and where it’s negative, you can test one value to the left of $x_{i}$ and one value to the right of $x_{i}$ by inserting it for $x$ in $(x - x_{i})$ . When the answer is positive you draw a solid line, and when it’s negative you draw a dashed line. Repeat this for all the factors and draw the sign charts underneath each other.
7.: In the end, you can sum the signs of all the sign charts you’ve drawn to make a sign chart representing the entire function below the charts of all the factors.

Example 3

Find the zeros of the function

f (x) = x^{2} - 7 x + 12

and decide where the graph is above or below the $x$ -axis

1.

First you need to solve the equation

f (x) = 0

. In this case, you can use the quadratic formula or inspection. You find the zeros

x_{1} = 4

and

x_{2} = 3

2.

That means the factorization of the function is

(x - 4) (x - 3) .

3.

Draw the sign chart for

(x - 3)

. First, you find where

x - 3 = 0

\begin{array}{l} x - 3 & = 0 \\ x & = 3 \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.

4.

Draw the sign chart for

(x - 4)

. First, you find where

x - 4 = 0

\begin{array}{l} x - 4 & = 0 \\ x & = 4 \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 $(x - 3)$ .

5.

Add the two sign charts together and draw the resulting sign chart below them:

The sign chart of (x-3)(x-4) is found by combining the sign charts of x-3 and x-4

6.

Conclude by stating the intervals you were looking for. The graph is above the

x

-axis on the interval

(- \infty, 3) \cup (4, \infty),

and it’s below the $x$ -axis on the interval $(3, 4)$ . When $x = 3$ and $x = 4$ , the graph is neither above or below the $x$ -axis.

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