AI Tutor
If a company's profit flips from -\50{,}000toto+$50{,}000$, its magnitude stays the same while its direction reverses. What happens geometrically when you wrap an entire function inside an absolute value sign versus placing it just around the input variable?
The modulus (or absolute value) of a real number aa, written ∣a∣|a|, is its non-negative distance from zero on a number line, so βˆ£βˆ’5∣=5|{-5}| = 5 and ∣5∣=5|5| = 5. In 1841, German mathematician Karl Weierstrass introduced the vertical bars notation ∣x∣|x| to standardize this fundamental concept of magnitude.
How do we systematically sketch the transformation when the modulus bars wrap around the entire function?

The Transformation y = |f(x)|

In the equation y=∣f(x)∣y = |f(x)|, the modulus acts on the final output value of the function.
Any portion of y=f(x)y = f(x) where yβ‰₯0y \ge 0 remains completely unchanged, while any portion where y<0y < 0 is reflected vertically across the x-axis into positive territory.
The points where the curve meets the xx-axis are called invariant points because their yy-value is 00, meaning ∣0∣=0|0| = 0. At these points, the reflected curve forms sharp V-shaped corners called cusps.
What happens if we shift the modulus inside the brackets so it acts directly on the input variable instead?

The Transformation y = f(|x|)

In the equation y=f(∣x∣)y = f(|x|), every negative input is turned positive before the function evaluates it, so an input of x=βˆ’3x = -3 outputs f(βˆ£βˆ’3∣)=f(3)f(|{-3}|) = f(3).
To sketch y=f(∣x∣)y = f(|x|), delete the original curve for x<0x < 0, keep the curve for xβ‰₯0x \ge 0, and reflect this right-hand half horizontally across the y-axis.