Twelfth Grade · Math · week 2

Transformations of functions, fully combined

Apply shifts, both reflections, and stretches to any parent function in combination, and distinguish |f(x)| from f(|x|).

Algebra II taught you to shift, reflect over the x-axis, and stretch a graph. Precalculus adds the transformation you have been missing -- reflecting over the y-axis -- and asks you to combine every move fluently, on parent functions beyond the parabola: sqrt(x), x^3, 1/x, and |x|.

The new move: g(x) = f(-x) reflects the graph over the y-axis, flipping it left-right by negating the INPUT before f ever sees it. This differs from -f(x), which negates the OUTPUT after f is done. Keep them straight by asking which side of the function the negative sign sits on.

A second easily-confused pair: |f(x)| takes the ordinary graph of f and flips any part BELOW the x-axis up above it (the output becomes nonnegative). f(|x|) instead erases the left half of the graph (x<0) and replaces it with a mirror image of the right half (the input becomes nonnegative first). These generally produce two different graphs from the same f.

📋 For the teacher
Teach it (10–15 min):
  • Demonstrate f(-x) with an asymmetric graph (sqrt(x) works well) by physically folding a printed copy over the y-axis -- seeing the fold beats memorizing the rule.
  • Sketch |f(x)| and f(|x|) side by side for the SAME f, using an asymmetric function, so the difference is visual, not just verbal.
  • For a stack of transformations, check one point by direct substitution every single time, exactly like the example -- do not trust a memorized order when a reflection and a shift both touch x.
  • Reconnect to last week's even/odd work out loud: an even function is unchanged by a y-axis flip, and an odd function is unchanged by a combined double flip -- transformations and symmetry are the same idea seen from two directions.
Talk about it:
  • Why does f(-x) reflect a graph over the y-axis instead of the x-axis?
  • Find a value of x where |f(x)| and f(|x|) give the SAME output, for f(x) = x - 5. What has to be true for that to happen?
  • Could a transformation ever turn an odd function into an even one? What would it take?

Mastery looks like: They describe any combination of shifts, both reflections, and a vertical stretch in the correct order, verify it with a test point, and can distinguish |f(x)| from f(|x|) on a specific example.

Common stumbles: Treating f(-x) and -f(x) as the same move; assuming |f(x)| and f(|x|) always produce the same graph.

See it visually

Interactive, spoken-aloud tools that show this idea in motion.

AdditionSee groups combine — with counters, blocks, and the number line.try →
Homework for this lessonType it, write it with a stylus, or print it for pencil & paper.open →
Records attendance + 15 min of Math for this learner.
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