AI Tutor
When you turn the volume knob on a stereo, you don't change the tempo of the music — you change the amplitude of the sound wave, pulling peaks higher and troughs lower.
In mathematics, placing an outside multiplier on a function does the exact same thing to its graph.

The Outside Multiplier

When we write g(x)=a⋅f(x)g(x) = a \cdot f(x), the constant aa acts directly on the output values (yy-values) after the function rule has been evaluated.
Every point (x,y)(x, y) on the parent graph transforms directly into the new point (x,a⋅y)(x, a \cdot y).
How does the magnitude of this multiplier change whether the graph pulls taller or squashes flatter?

Stretches vs. Compressions

If ∣a∣>1|a| > 1, the function undergoes a vertical stretch by a factor of ∣a∣|a|, pulling all points away from the xx-axis.
If 0<∣a∣<10 < |a| < 1, the function undergoes a vertical compression (or shrink) by a factor of ∣a∣|a|, pushing all points closer to the xx-axis.
Notice that points lying directly on the xx-axis where y=0y = 0 remain completely unchanged, because a⋅0=0a \cdot 0 = 0.
What happens if we apply these exact coordinate rules to a specific function step by step?