Question

f(x) = 3x^2
f(x) = 0.3x^2

Answer 1

You’ve given two quadratic functions. Here are the key comparisons and properties.

Functions
- f1(x) = 3x^2
- f2(x) = 0.3x^2

Relationship
- f2(x) = 0.1 · f1(x). So f2 is exactly one tenth of f1 for every x.
- Geometrically, f2 is a vertical compression (shrink) of f1 by a factor of 10 about the x-axis.

Common properties
- Domain: all real numbers (−∞, ∞) for both.
- Range: [0, ∞) for both (since x^2 ≥ 0 and coefficients are positive).
- Vertex: both have vertex at (0, 0).
- Axis of symmetry: x = 0 for both.
- Both open upward (positive leading coefficient).

Derivatives and integrals
- f1′(x) = 6x, ∫f1(x) dx = x^3 + C.
- f2′(x) = 0.6x, ∫f2(x) dx = 0.1 x^3 + C.
- Derivatives (slopes) and antiderivatives are scaled by the same factor 0.1 between the two functions.

Example values
- x = 1: f1 = 3, f2 = 0.3
- x = 2: f1 = 12, f2 = 2.4
- x = 0.5: f1 = 0.75, f2 = 0.075

If you want, I can:
- Plot both graphs,
- Solve equations like f(x)=a,
- Show transformations step-by-step,
- Or do any calculus (max/min, integrals between bounds, etc.). Which would you like?