Graphing absolute value functions.
Find the corner, then the arms.

Plot four functions in y=a|x-h|+k. Identify each vertex and symmetry line, calculate sample values and compare upward, downward and wider V shapes.

Choose Letter or A4, then download a student sheet first. Keep the separate reference or answer file closed until you have attempted the case. Every worksheet, SVG and answer is free without signup.

All examples use original fictional data and coordinate units. They are practice material, not measured classroom results, published questions or evidence about real people.

AV01: A centered V

The first case shows the checking pattern. Individual answer files contain the completed graph and result for every case.

1. Locating the Vertex Precisely

   Begin by identifying the vertex coordinates h and k from the function definition y equals a absolute value of x minus h plus k. For the centered V case, the vertex is at zero zero. For the shifted V, it is at two negative one. Mark this point clearly on your grid. Do not rely on physical ruler measurements to find this spot. Instead, use the listed coordinates provided in the reference PDF. This ensures the sharp corner aligns exactly with the mathematical definition. The vertex serves as the anchor for all subsequent plotting steps. Accurate placement here prevents distortion in the rest of the graph. Verify the coordinates against the supplied values before proceeding to the next step.

2. Plotting Symmetric Input Values

   Use the table of symmetric inputs around the vertex h to calculate y values. For the reflected V case with vertex at negative one three, use inputs negative four, negative two, negative one, zero, and two. Calculate each output using the function rule. Plot these points on the coordinate plane. Connect the points with straight lines to form the V shape. Ensure the lines meet at a sharp corner at the vertex. Do not smooth the corner. The symmetry of the inputs ensures the graph balances correctly around the vertical axis of symmetry. Check your plotted points against the answer PDF to confirm accuracy. This method relies on calculation, not physical estimation.

3. Applying the Scale Factor

   Adjust the width of the V by applying the coefficient a. For the wider V case, use a equals one half. This value stretches the graph vertically compared to the standard V. Calculate the y values using this factor. The vertex remains at one negative two. The inputs are negative two, zero, one, two, and four. Plot these calculated points. The graph will appear broader than the standard case. Do not use a physical ruler to estimate the width. Rely on the calculated coordinates. The negative sign in other cases flips the graph downward. Verify the shape matches the reference image. The scale factor directly controls the steepness of the lines.

Choose the PDF that matches your paper. Print at Actual size / 100% and turn off Fit, Shrink and browser headers. SVG is editable but an editor can rescale it; the PDF is the simpler print route. Physical printer, classroom and curriculum tests have not been completed.

Further reading: OpenStax: absolute value functions. The diagrams, datasets and exercises here are original Zumola work.