You already know each move on its own (Shifting and Stretching Graphs): $f(x) + k$ shifts up, $f(x - h)$ shifts right, $a f(x)$ stretches vertically and $f(-x)$ reflects across the $y$-axis. Now you will combine them into one formula and keep track of the order.
Key idea
In $g(x) = a\,f\big(b(x - h)\big) + k$, the outside numbers act on outputs: multiply by $a$, then add $k$. The inside numbers act on inputs, backwards: divide by $b$, then add $h$. A point $(x, y)$ on $f$ lands at $\left(\frac{x}{b} + h,\ ay + k\right)$.
Why it works
Follow one point. Outside, $g$ computes $f(\dots)$, multiplies by $a$ and adds $k$, in that order, just like the order of operations. Inside, $g$ needs to feed $f$ the same input as before. If $f$ had the point $(4, 2)$, then $g$ reaches that height when $b(x - h) = 4$. Solving for $x$ undoes the inside steps in reverse: $x = \frac{4}{b} + h$. That is why inside changes feel backwards.
Worked examples
Worked example: Reading a combined transformation
Describe how $g(x) = -2(x - 3)^2 + 1$ comes from $f(x) = x^2$, and find where the point $(1, 1)$ lands.
- Inside: $x - 3$ shifts the graph 3 to the right.
- Outside: $-2$ reflects across the $x$-axis and stretches by 2; then $+1$ shifts up 1.
- The point $(1, 1)$: the new $x$ solves $x - 3 = 1$, so $x = 4$. The new $y$ is $-2 \cdot 1 + 1 = -1$.
- Check: $g(4) = -2(1)^2 + 1 = -1$.
Answer: Reflect, stretch by 2, right 3, up 1; $(1, 1)$ lands at $(4, -1)$.
Worked example: Factor the inside first
Where does the graph of $g(x) = \sqrt{2x - 6}$ start, and where does the point $(4, 2)$ of $\sqrt{x}$ land?
- Factor out the 2: $2x - 6 = 2(x - 3)$. So $b = 2$ (squeeze toward the $y$-axis by $\frac{1}{2}$) and $h = 3$.
- The parent starts at $(0, 0)$. The new start solves $2(x - 3) = 0$: $x = 3$. So $g$ starts at $(3, 0)$.
- For $(4, 2)$: solve $2(x - 3) = 4$, so $x - 3 = 2$ and $x = 5$. The height stays 2.
- Check: $g(5) = \sqrt{10 - 6} = 2$.
Answer: Starts at $(3, 0)$; $(4, 2)$ lands at $(5, 2)$.
Worked example: Writing the equation
Reflect $y = |x|$ across the $x$-axis, stretch it vertically by 3, then shift it 2 left and 4 down.
- Reflect and stretch: $y = -3|x|$.
- Left 2 means replace $x$ with $x + 2$: $y = -3|x + 2|$.
- Down 4: $y = -3|x + 2| - 4$. The vertex moved from $(0, 0)$ to $(-2, -4)$.
Answer: $y = -3|x + 2| - 4$
Common mistake
Reading $\sqrt{2x - 6}$ as a shift of 6. The 2 multiplies the whole inside, so factor it out: $2x - 6 = 2(x - 3)$, a shift of 3. Quick check: find where the inside equals 0. For $2x - 6 = 0$ that is $x = 3$, so the key point moved to $x = 3$.