Inequality Grapher

About the Inequality Grapher

Plot a function and visually identify where it is positive (above the x-axis) or negative.

How to use this calculator

1. Enter a function, such as \(f(x)=x^2-4\), into the grapher.
2. Choose whether you want the graph to show positive regions, negative regions, or both.
3. Look for where the curve crosses the x-axis, because those points usually separate the sign intervals.
4. Read the intervals from the graph, then write the answer using interval notation or inequality notation.

The formula explained

The graph shows where \(f(x)\) is positive, meaning \(f(x)>0\), and where it is negative, meaning \(f(x)<0\). The x-axis crossings are the boundary points where the sign can change.

• x = the input value on the horizontal axis
• f(x) = the output of the function being graphed
• 0 = the x-axis level used to test whether the graph is positive or negative

Step by step method

1. Set the function equal to \(0\) to find the x-intercepts, because those are the points where the sign can change.
2. Test a point in each interval between the intercepts to see whether \(f(x)\) is positive or negative.
3. Use the graph to confirm which intervals are above the x-axis and which are below it.

Worked example

Problem. Graph \(f(x)=x^2-4\) and identify where the function is positive and where it is negative.

1. Find the x-intercepts by solving \(x^2-4=0\), which gives \(x=-2\) and \(x=2\).
2. Check the intervals \((-infty,-2)\), \((-2,2)\), and \((2,infty)\). For \(x=0\), \(f(0)=-4\), so the graph is below the x-axis between \(-2\) and \(2\). For \(x=3\), \(f(3)=5\), so the graph is above the x-axis to the right of \(2\).
3. Therefore, \(f(x)>0\) on \((-infty,-2)\cup(2,infty)\) and \(f(x)<0\) on \((-2,2)\).

Tips and common mistakes

• Do not include the x-intercepts when the inequality is strict, like \(>0\) or \(<0\), because the function equals \(0\) there.
• If the graph only touches the x-axis and turns around, the sign may not change at that point.

Frequently asked questions