Absolute value graphs

Lesson 2 of 4 · about 15 minutes · Win: look at y = a|x − p| + q and instantly say its shape and salient point.

1. The three-step method from your notes

Every graph example in your notes (abs.pdf pp.5–7) uses the same three steps: 1. Form, 2. Salient point, 3. Draw the graph. The salient point is the sharp corner of the V.

Form y = a|x| + q (abs.pdf p.5): salient point (0; q).
Form y = a|x − p| (abs.pdf p.6): salient point (p; 0), because the inside is zero when x = p.
General y = a|x − p| + q (abs.pdf p.5 heading): salient point (p; q).
Shape: a > 0 → “V”; a < 0 → “⋀”. Then draw two straight lines from the salient point.
Domain D_f = ℝ; range of y = |x| is y ≥ 0 (abs.pdf p.3).

Sign trap: |x + 1| = |x − (−1)|, so p = −1, not +1. The x-coordinate of the salient point is the value that makes the inside zero.

Tip for drawing: plot the salient point, then one more point (e.g. the y-intercept, put x = 0), and use symmetry.

2. Your notes’ examples, sketched

Example	Form, salient point, shape	Sketch

3. Practice

Read / watch

Primary: abs.pdf pp.3–7. External: Khan Academy – Absolute value equations & functions (see the “graphs of absolute value functions” videos; Khan writes the salient point as the “vertex”).

Unsure how to find the equation from a graph (abs.pdf p.7)? Ask your teacher-agent: read off (p; q), then use one more point to find a.

Next lesson: which factoring rule? →