Revise: Elementary Functions and Transformations

y=x³: S-shaped, passes through origin, odd function.

x=3 → y=27.

y=a/x: two branches, asymptotes at both axes.

y=6/x, x=3 → y=2.

y=a·bˣ: growth if b>1, decay if 0<b<1, asymptote y=0.

y=3·2ˣ, x=4 → y=48.

y=f(x−h)+k: shift h horizontally, k vertically.

4 left, 5 down → y=(x+4)²−5.

y=a·f(x): vertical scale. y=f(bx): horizontal, inverted.

f(2x) compresses horizontally.

Recognize linear, quadratic, cubic, reciprocal, sqrt, exponential shapes.

y=5/x → reciprocal family.

y=x³ is an odd function: f(−x)=−f(x).

x=−2 → y=−8.

Undefined at x=0; models inverse proportion.

Speed and time for fixed distance.

y=a√(x−h)+k; domain x≥h.

√(x−3), x=12 → y=3, domain x≥3.

y-intercept is always a, since bˣ=1 at x=0.

y=5·2ˣ, x=0 → y=5.

(x−h) shifts right; (x+h) shifts left — feels backwards.

3 right, 2 up on √x → √(x−3)+2.

y=−f(x): flip over x-axis. y=f(−x): flip over y-axis.

x³ reflected over y-axis → −x³.

Horizontal scaling is inverted: |b|>1 compresses.

(4,5) on f(x) → (2,5) on f(2x).

y=a·f(b(x−h))+k: identify a,b,h,k separately.

−2(x+1)²+4: left 1, stretch×2, flip, up 4.