Unit 2 · Topic 1.12
Transformations of Functions
Construct transformed functions, map every point systematically, and track how translations, dilations, and reflections change domain, range, and key graph features.
1. Preimage and Image
The original graph of \(f\) is the preimage. After a transformation, the resulting graph of \(g\) is the image.
A transformation acts on every point of the graph according to one consistent rule. Tracking a few key points is often more reliable than trying to redraw the shape from memory.
2. Master Transformation Table
| New function | Effect on graph | Point mapping from \((x,y)\) |
|---|
| \(f(x)+k\) | Vertical translation by \(k\) | \((x,y+k)\) |
| \(f(x-h)\) | Horizontal translation right by \(h\) | \((x+h,y)\) |
| \(a f(x)\) | Vertical dilation by \(|a|\); reflect over x-axis if \(a<0\) | \((x,ay)\) |
| \(f(bx)\) | Horizontal dilation by \(1/|b|\); reflect over y-axis if \(b<0\) | \((x/b,y)\) |
Outside operations change outputs directly. Inside operations change which input reaches the original function, so horizontal signs and scale factors appear reversed or reciprocal.
3. Vertical Translations
\[g(x)=f(x)+k.\]
- If \(k>0\), shift the graph up \(k\) units.
- If \(k<0\), shift the graph down \(|k|\) units.
- Every output changes by \(k\); inputs stay fixed.
A vertical translation shifts the range and every horizontal asymptote by \(k\), while preserving the domain.
4. Horizontal Translations
\[g(x)=f(x-h).\]
- If \(h>0\), shift right \(h\) units.
- If \(h<0\), shift left \(|h|\) units.
- Each original input \(u\) appears at the new input \(u+h\).
The direction looks opposite to the sign inside because \(x-h=u\) gives \(x=u+h\).
5. Vertical Dilations and Reflections
\[g(x)=a f(x),\qquad a\ne0.\]
- \(|a|>1\): vertical stretch.
- \(0<|a|<1\): vertical compression.
- \(a<0\): reflection across the x-axis in addition to the dilation.
Inputs remain fixed while every output is multiplied by \(a\).
6. Horizontal Dilations and Reflections
\[g(x)=f(bx),\qquad b\ne0.\]
- \(|b|>1\): horizontal compression by \(1/|b|\).
- \(0<|b|<1\): horizontal stretch by \(1/|b|\).
- \(b<0\): reflection across the y-axis in addition to the dilation.
To reproduce the original input \(u\), solve \(bx=u\), so the new input is \(u/b\).
7. General Point-Mapping Formula
For the combined transformation
\[g(x)=a f(b(x-h))+k,\qquad a\ne0,\ b\ne0,\]
let \((u,v)\) be a point on \(f\), so \(v=f(u)\). Set the new inside input equal to \(u\):
\[b(x-h)=u\quad\Longrightarrow\quad x=h+\frac{u}{b}.\]
The new output is \(av+k\). Therefore
\[\boxed{(u,v)\mapsto\left(h+\frac{u}{b},\ k+av\right)}.\]
8. Worked Combined Transformation
Let
\[g(x)=-2f(3(x-4))+5.\]
Here \(a=-2\), \(b=3\), \(h=4\), and \(k=5\). The graph is horizontally compressed by \(1/3\), shifted right 4, reflected across the x-axis, stretched vertically by 2, and shifted up 5.
| Point on \(f\) | New input \(4+u/3\) | New output \(5-2v\) | Point on \(g\) |
|---|
| \((-3,2)\) | 3 | 1 | \((3,1)\) |
| \((0,-1)\) | 4 | 7 | \((4,7)\) |
| \((6,4)\) | 6 | \(-3\) | \((6,-3)\) |
9. Factor the Inside Expression First
The expression \(f(2x+6)\) should be rewritten as
\[f(2x+6)=f(2(x+3)).\]
This reveals a horizontal compression by \(1/2\) and a shift left 3. Reading the un-factored constant 6 as a six-unit shift is incorrect.
10. Transform Domain and Range
For \(g(x)=a f(b(x-h))+k\), transform domain inputs by \(u\mapsto h+u/b\) and range outputs by \(v\mapsto k+av\).
Start with \(f(x)=\sqrt{x}\), whose domain and range are both \([0,\infty)\). Let
\[g(x)=-3\sqrt{2(x-1)}+4.\]
- Domain: \(2(x-1)\ge0\), so \(x\ge1\).
- Range: original outputs satisfy \(v\ge0\); \(-3v+4\le4\), so \(y\le4\).
- The original endpoint \((0,0)\) maps to \((1,4)\).
11. Key Features Follow the Same Mapping
| Feature on \(f\) | Corresponding feature on \(g(x)=a f(b(x-h))+k\) |
|---|
| Point or extremum \((u,v)\) | \((h+u/b,\ k+av)\) |
| Vertical asymptote \(x=u\) | \(x=h+u/b\) |
| Horizontal asymptote \(y=v\) | \(y=k+av\) |
| Hole \((u,v)\) | Hole at \((h+u/b,\ k+av)\) |
| Domain interval | Map both endpoints with \(u\mapsto h+u/b\); reverse order if \(b<0\) |
| Range interval | Map outputs with \(v\mapsto k+av\); reverse order if \(a<0\) |
A zero of \(f\) does not necessarily remain a zero of \(g\) when \(k\ne0\), because vertical translation changes the target output.
12. Transform a Table Without a Formula
If a table gives \(f(u)=v\), use the point-mapping formula directly. For \(g(x)=2f(4(x+1))-3\),
\[(u,v)\mapsto\left(-1+\frac{u}{4},\ 2v-3\right).\]
This method works even when no equation for \(f\) is available. Keep each input paired with its original output before transforming both coordinates.
13. Construct a Transformation from a Description
Suppose a model must run twice as fast horizontally, be reflected across the x-axis, have outputs tripled in magnitude, then shift right 2 and up 7.
\[g(x)=-3f(2(x-2))+7.\]
Translate each phrase into one parameter, then test a known point to confirm the formula. In context, explain how input units and output units are changed by the dilations and translations.
14. Technology, Common Errors, and AP Reasoning
- Graph the parent and transformed functions together and verify several mapped points.
- Do not read an inside sign in the same direction as an outside shift.
- Use the reciprocal \(1/|b|\) for horizontal dilation.
- Factor the complete inside expression before identifying \(h\).
- Do not confuse reflection across the x-axis with reflection across the y-axis.
- Apply \(a\) before adding \(k\) to outputs.
- Reverse interval endpoint order when a negative factor reflects a domain or range.
- Do not assume domain and range remain unchanged.
Complete response pattern: identify \(a,b,h,k\), state each transformation, map a key point, and verify the resulting domain, range, or graph feature.
Checkpoint · Topic 1.12
Let \(g(x)=\frac{1}{2}f(-2(x+3))-4\). The graph of \(f\) contains \((-4,6)\), \((0,-2)\), and \((8,4)\), has domain \([-4,8]\), range \([-2,6]\), and a vertical asymptote at \(x=2\).
- Identify \(a,b,h,k\) and describe every transformation.
- Map all three given points to the graph of \(g\).
- Determine the transformed domain and range, including correct endpoint order.
- Find the transformed vertical asymptote.
- Write a point-mapping rule and use it to check every answer.
- Explain why the negative inside factor and the positive outside factor affect different coordinates.