Topic A: Transformations of Functions
Lessons 1–5 · Shifts, reflections, and stretches.
01–03 — Lessons 1–3
Vertical & Horizontal Transformations
- f(x) + k: shift up k. f(x) − k: shift down k.
- f(x − h): shift right h. f(x + h): shift left h.
- −f(x): reflect over x-axis. f(−x): reflect over y-axis.
- af(x), a > 1: vertical stretch. 0 < a < 1: vertical compression.
- Apply in this order: horizontal shift → reflection → stretch/compression → vertical shift.
g(x) = −2f(x − 3) + 1: shift right 3, reflect over x-axis, stretch by 2, shift up 1.
Practice — Lessons 1–3
Q1 · Multiple Choice
f(x) is shifted 4 units right. The new function is:
Q2 · Multiple Choice
How does g(x) = −f(x) relate to f(x)?
04–05 — Lessons 4–5
Stretches & Compressions
- Vertical stretch: a·f(x), a > 1 — pulls graph away from x-axis.
- Vertical compression: a·f(x), 0 < a < 1 — pushes toward x-axis.
- Horizontal stretch: f(bx), 0 < b < 1.
- Horizontal compression: f(bx), b > 1.
- Stretches and compressions change shape, not just position.
g(x) = 3f(x) → every y-value is multiplied by 3 (vertical stretch by factor 3).
Practice — Lessons 4–5
Q3 · Multiple Choice
g(x) = (1/2)f(x). Compared to f(x), g(x) is:
Q4 · Multiple Choice
If f(2) = 6, what is 3f(2)?
Topic B: Multiple Representations & Function Families
Lessons 6–10 · Connecting graphs, tables, equations, and context.
06–10 — Lessons 6–10
Identifying & Comparing Function Families
Match real-world situations and representations to the right function type.
- Linear: constant rate of change — straight line, y = mx + b.
- Quadratic: rate of change is changing — parabola, y = ax² + bx + c.
- Exponential: multiplied by constant factor — rapid growth/decay, y = abˣ.
- Absolute value: V-shape, y = |x − h| + k.
- Use tables: if differences are constant → linear. If ratios are constant → exponential.
Table: x=0,1,2,3 → y=1,2,4,8. Each y doubles → exponential: y = 2ˣ.
Practice — Lessons 6–10
Q5 · Multiple Choice
A table shows x: 0,1,2,3 and y: 5,8,11,14. What type of function?
Q6 · Multiple Choice
Which situation is best modeled by an exponential function?