Question

Specify the Key differences between the given Quadratic functions to a Quadratic Parent Function
f(x) = 3x^2
f(x) = 0.3x^2

Answer 1

Compare each function to the quadratic parent f(x) = x^2 (where the leading coefficient a = 1).

Common features (parent and both given functions)
- Vertex at (0, 0).
- Axis of symmetry: x = 0.
- Open upward (concave up) because the leading coefficient is positive.
- Domain: all real numbers (−∞, ∞).
- Range: y ≥ 0.
- y‑intercept and only x‑intercept: (0, 0).

Differences (caused by the leading coefficient a)
1. f(x) = 3x^2 (a = 3)
- Vertical stretch by factor 3 (|a| > 1): the parabola is narrower/steeper than the parent.
- Every y-value is 3 times the parent’s y-value: f(x) = 3·(x^2).
- Example values: f(1)=3 (vs 1), f(2)=12 (vs 4).
- Derivative: f′(x) = 6x (3 times the parent derivative 2x).

2. f(x) = 0.3x^2 (a = 0.3)
- Vertical compression by factor 0.3 (0 < |a| < 1): the parabola is wider/flatter than the parent.
- Every y-value is 0.3 times the parent’s y-value: f(x) = 0.3·(x^2).
- Example values: f(1)=0.3 (vs 1), f(2)=1.2 (vs 4).
- Derivative: f′(x) = 0.6x (0.3 times the parent derivative).

Short summary
- 3x^2: same shape shifted vertically (scaled) to be steeper (vertical stretch).
- 0.3x^2: same shape scaled to be wider (vertical compression).
- Neither is reflected or translated; both keep the parent’s vertex, axis, domain, and range.