The template \(y=a\,f(b(x-h))+k\)

Key idea

Many “advanced function transformations” problems use the standard form \[ y=a\,f(b(x-h))+k. \] This form tells you how the graph of \(y=f(x)\) changes:

• \(a\) controls vertical stretch/compression by \(|a|\) and reflects across the \(x\)-axis if \(a<0\).
• \(k\) controls the vertical shift: up \(k\), down if \(k<0\).
• \(b\) controls horizontal stretch/compression by \(\tfrac{1}{|b|}\) and reflects across the \(y\)-axis if \(b<0\).
• \(h\) controls the horizontal shift: right \(h\), left if \(h<0\).

The point-mapping rule (fast + reliable)

If \((x,y)\) is a point on \(y=f(x)\), then the corresponding point on \[ y=a\,f(b(x-h))+k \] is \[ \left(\frac{x}{b}+h,\; ay+k\right)\quad (b\neq 0). \] This is one of the quickest ways to avoid sign mistakes on “inside” transformations.

Worked example

Try it

Summary

• Standard form: \(y=a\,f(b(x-h))+k\).
• Point mapping: \((x,y)\mapsto\left(\tfrac{x}{b}+h,\;ay+k\right)\) (for b≠ 0).

Vertical shifts, stretches, and reflections: \(y=a\,f(x)+k\)

Key idea

• Vertical shift: \(y=f(x)+k\) shifts up \(k\); \(y=f(x)-k\) shifts down \(k\).
• Vertical stretch/compression: \(y=a\,f(x)\) multiplies all \(y\)-values by \(a\). If \(|a|>1\), it stretches; if \(0<|a|<1\), it compresses.
• Reflection across the \(x\)-axis: \(y=-f(x)\) flips the graph vertically.

Worked example

Try it

Summary

• Outside changes affect \(y\)-values: \(y=a\,f(x)+k\).
• Negative outside \(f\) reflects across the \(x\)-axis.

Function notation and inside transformations

Key idea

When you see \(f(\,\text{something}\,)\), you substitute the “something” wherever \(x\) appears in the rule for \(f(x)\). Then you interpret the graph change:

• \(y=f(x-h)\): shift right \(h\).
• \(y=f(x+h)\): shift left \(h\).
• \(y=f(bx)\): horizontal scale by \(\tfrac{1}{|b|}\) (compression if \(|b|>1\), stretch if \(0<|b|<1\)).
• \(y=f(-x)\): reflection across the \(y\)-axis.

Worked example

Try it

Summary

• Inside changes affect \(x\)-values (horizontal shifts/scales/reflections).
• Always substitute first, then simplify.

Composite transformations and the correct order

Key idea

When a transformation includes both inside and outside changes, use the template: \[ y=a\,f(b(x-h))+k. \] A reliable transformation order (from \(y=f(x)\) to the new graph) is:

• 1) Horizontal scale / reflection: based on \(b\) (scale by \(\tfrac{1}{|b|}\); reflect across \(y\)-axis if \(b<0\)).
• 2) Horizontal shift: based on \(h\) (right \(h\)).
• 3) Vertical scale / reflection: based on \(a\) (scale by \(|a|\); reflect across \(x\)-axis if \(a<0\)).
• 4) Vertical shift: based on \(k\) (up \(k\)).

Worked example

Try it

Summary

• Use \(y=a\,f(b(x-h))+k\) to read composite transformations.
• Horizontal scaling (from \(b\)) comes before horizontal shifting (from \(h\)).

Transformations of \(|x|\), \(\sqrt{x}\), \(a^x\), and \(\sin(x)\)

Key idea

• Absolute value \(y=|x|\): key feature is the vertex at \((0,0)\).
• Square root \(y=\sqrt{x}\): domain starts at \(x\ge 0\). Transformations move the “starting point” and can change the domain.
• Exponential \(y=a^x\) (\(a>0\), a≠ 1): has a horizontal asymptote at \(y=0\) (before vertical shifts).
• Sine \(y=\sin(x)\): for \(y=A\sin(B(x-h))+k\), amplitude is \(|A|\) and period is \(\dfrac{2\pi}{|B|}\).

Worked example

Try it

Summary

• Transformations apply to all parent functions, but keep an eye on key features (vertex, asymptotes, period) and domain restrictions.

Reverse-engineer the steps (and build the equation)

Reliable strategy

• 1) Rewrite the inside as \(b(x-h)\) when possible.
• 2) Identify \(a,b,h,k\) from \(y=a\,f(b(x-h))+k\).
• 3) List transformations in the order: horizontal scale/reflect → horizontal shift → vertical scale/reflect → vertical shift.
• 4) Check by mapping one easy point (or a key feature).

Worked example

Try it

Summary

• Rewrite → identify \(a,b,h,k\) → list transformations in a consistent order → check with a mapped point.

Why advanced function transformations matter

Where function transformations show up

• Fast graphing: sketch complex functions by transforming a parent graph instead of plotting many points.
• Modeling: fit data by shifting/scaling a known shape (exponentials, trigonometric curves, absolute value “V” shapes).
• Solving equations: understand how shifts and scales move intercepts and intersections.
• Precalculus & calculus: transformations help you reason about domains, ranges, asymptotes, and behavior without heavy computation.

Worked example: transform a parent function

Try it

Final recap

• Standard form: \(y=a\,f(b(x-h))+k\).
• Horizontal: scale by \(\tfrac{1}{|b|}\), reflect across \(y\)-axis if \(b<0\), then shift right by \(h\).
• Vertical: scale by \(|a|\), reflect across \(x\)-axis if \(a<0\), then shift up by \(k\).
• Substitution skill: compute expressions like \(f(x+1)\), \(f(x-4)\), \(f(-x+1)\), and \(f(0.5x)\) by replacing \(x\).
• Point mapping (fast check): \((x,y)\mapsto\left(\tfrac{x}{b}+h,\;ay+k\right)\) for \(y=a\,f(b(x-h))+k\).