Absolute Value Grapher

About the Absolute Value Grapher

Plot y = |f(x)| and see the characteristic V-shape and reflections.

How to use this calculator

1. Enter a function \(f(x)\) such as \(x^2-4x-5\).
2. Choose the graphing range if your tool lets you adjust it.
3. Plot \(y=|f(x)|\) to see the reflected graph.
4. Compare the original function and the absolute value graph to identify which sections were moved above the x-axis.

The formula explained

The graph of \(y=|f(x)|\) shows the absolute value of each output of \(f(x)\). This means negative values become positive, so the graph reflects any below-axis parts upward.

• x = the input value on the horizontal axis
• f(x) = the original function value before absolute value is applied
• |f(x)| = the absolute value of the output, always zero or positive

Step by step method

1. Graph the original function \(f(x)\) or identify a few points from it.
2. For each point with a negative \(y\)-value, change it to its positive opposite, because absolute value makes outputs nonnegative.
3. Keep every point with a nonnegative \(y\)-value unchanged.
4. Plot the new points and connect them, noting that any reflected part may meet the x-axis in a sharp corner.

Worked example

Problem. Graph \(y=|x^2-4x-5|\) and describe what happens to the part of the graph below the x-axis.

1. First find where \(x^2-4x-5=0\). Factoring gives \((x-5)(x+1)=0\), so the x-intercepts are \(x=-1\) and \(x=5\).
2. Check a point between them, such as \(x=0\): \(0^2-4(0)-5=-5\), so that part of the graph is below the x-axis and will be reflected upward.
3. After applying absolute value, the point \((0,-5)\) becomes \((0,5)\), while points like \((6,7)\) stay the same. The graph of \(y=|x^2-4x-5|\) is the original parabola with its negative part flipped above the x-axis.

Tips and common mistakes

• Only the output values get an absolute value, not the input \(x\). So \(|f(x)|\) is different from \(f(|x|)\).
• If the original graph touches or crosses the x-axis, those points stay on the x-axis after the reflection, so they often mark the sharp corners of the new graph.

Frequently asked questions