Section 1.comp

Composition & Transformations

Composition and transformations are among the most-tested topics on the final — expect 3–4 questions. Every time you chain calculations or modify a known graph, you're using these skills.

⚡ Quick Summary

Composition is just stacking two machines together — the output of the first machine becomes the input of the second. If $g$ doubles a number and $f$ adds 1, then feeding 3 into $g$ first gives 6, and feeding 6 into $f$ gives 7. The order matters: reversing which machine goes first usually gives a different answer.

Average Rate of Change

\frac{f(b) - f(a)}{b - a}

Slope of the secant line between (a, f(a)) and (b, f(b))

Composition of Functions

(f \circ g)(x) = f(g(x))

Apply g first, then f — read right to left

📋 Before You Start

Tap any item if you need a refresher:

Plain-English version

The average rate of change is how fast something is changing between two points — it's the same idea as "miles per hour" for a road trip. You take the difference in outputs divided by the difference in inputs, which is just the slope of the line connecting two points on the graph.

Transformations let you take a simple, familiar graph (like a parabola or V-shape) and slide it around, stretch it, or flip it. Adding a number outside the function moves the graph up or down. Changing what's inside the function (like replacing $x$ with $x - 3$ ) shifts it left or right — but here's the tricky part: inside changes go the opposite direction from what you'd expect. Subtracting 3 inside actually moves the graph *right*.

The absolute value function makes a V-shape because it turns every negative input positive — it measures distance from zero. You can shift, stretch, and flip this V using the same transformation rules: the vertex (the tip of the V) tells you where the center of the shape landed after all the shifts.

Key Formulas

Average Rate of Change

\frac{f(b) - f(a)}{b - a}

Slope of the secant line between (a, f(a)) and (b, f(b))

Composition of Functions

(f \circ g)(x) = f(g(x))

Apply g first, then f — read right to left

Vertical & Horizontal Shifts

f(x) + k \;\text{(up/down)} \qquad f(x - h) \;\text{(right } h \text{)}

Inside changes move opposite: x - h shifts RIGHT h units

Stretches & Reflections

a \cdot f(x) \;\text{(vert. stretch)} \qquad f(bx) \;\text{(horiz. compress by } \tfrac{1}{b}\text{)}

-f(x) reflects over x-axis; f(-x) reflects over y-axis

Absolute Value (Vertex Form)

f(x) = a|x - h| + k, \quad \text{vertex } (h, k)

a > 0 opens up (V), a < 0 opens down (∧)

Key Takeaways

✓ $(f \circ g)(x) = f(g(x))$ — apply $g$ first, then $f$ ; order matters
✓Outside changes ( $+k$ , $a \cdot f$ ) affect $y$ directly; inside changes ( $x - h$ , $bx$ ) affect $x$ in the opposite way
✓ $f(x - h)$ shifts right $h$ ; $-f(x)$ reflects over the $x$ -axis; $f(-x)$ reflects over the $y$ -axis
✓Average rate of change $= \frac{f(b) - f(a)}{b - a}$ is just the slope between two points

⚠️ Common Mistakes

✗Applying composition in the wrong order — $(f \circ g)(x) = f(g(x))$ means apply $g$ first, then $f$ ; mixing this up gives a completely different answer
✗Shifting the graph the wrong direction — $f(x - 3)$ shifts RIGHT 3, not left; inside changes always go the opposite way from what you'd expect
✗Forgetting to distribute the negative in transformations — in $-2(x + 1)^2 + 5$ , the $-$ reflects over the $x$ -axis AND the $2$ stretches vertically; don't miss either piece
✗Mixing up average rate of change with instantaneous rate — average rate of change uses $\frac{f(b) - f(a)}{b - a}$ , which is the slope of the secant line between two points

Next: Section 1.inv Builds on: 1.fn

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1.fn Functions, Domain & Range

1.inv Inverse Functions