Pre-Calculus Function Transformations Cheat Sheet

Function transformations show how a graph changes when numbers are added, subtracted, multiplied, or placed inside a function rule. This cheat sheet helps students recognize shifts, reflections, stretches, and compressions without graphing every point. It is useful for comparing transformed functions to parent functions in algebra, pre-calculus, and graphing problems.

Understanding transformations makes it easier to write equations from graphs and predict graph behavior quickly.

The core idea is to compare a transformed function such as $g(x) = a f(b(x - h)) + k$ to its parent function $f(x)$ . The value $h$ moves the graph horizontally, $k$ moves it vertically, $a$ changes vertical size and reflection, and $b$ changes horizontal size and reflection. Transformations inside the input affect the graph horizontally and often work in the opposite direction students expect.

Transformations outside the function affect the graph vertically and usually work in the expected direction.

Key Facts

The general transformation form is $g(x) = a f(b(x - h)) + k$ , where $a$ , $b$ , $h$ , and $k$ control the graph.
The graph of $g(x) = f(x) + k$ shifts the parent graph vertically up $k$ units if $k > 0$ and down $|k|$ units if $k < 0$ .
The graph of $g(x) = f(x - h)$ shifts the parent graph right $h$ units if $h > 0$ and left $|h|$ units if $h < 0$ .
The graph of $g(x) = a f(x)$ is vertically stretched by factor $|a|$ if $|a| > 1$ and vertically compressed by factor $|a|$ if $0 < |a| < 1$ .
If $a < 0$ in $g(x) = a f(x)$ , the graph is reflected across the $x$ -axis.
The graph of $g(x) = f(bx)$ is horizontally compressed by factor $\frac{1}{|b|}$ if $|b| > 1$ and horizontally stretched by factor $\frac{1}{|b|}$ if $0 < |b| < 1$ .
If $b < 0$ in $g(x) = f(bx)$ , the graph is reflected across the $y$ -axis.
For $g(x) = a f(b(x - h)) + k$ , points on $f(x)$ transform from $(x,y)$ to $(h + \frac{x}{b}, ay + k)$ .

Vocabulary

Parent function: A parent function is the simplest function in a family, such as $f(x) = x^2$ , $f(x) = |x|$ , or $f(x) = \sqrt{x}$ .
Translation: A translation slides a graph horizontally or vertically without changing its shape.
Reflection: A reflection flips a graph across a line, such as the $x$ -axis for $g(x) = -f(x)$ or the $y$ -axis for $g(x) = f(-x)$ .
Vertical stretch or compression: A vertical stretch or compression multiplies all output values by $a$ in $g(x) = a f(x)$ .
Horizontal stretch or compression: A horizontal stretch or compression changes input values by $b$ in $g(x) = f(bx)$ , using the scale factor $\frac{1}{|b|}$ .
Transformation form: Transformation form is $g(x) = a f(b(x - h)) + k$ , which organizes vertical changes, horizontal changes, and reflections.

Common Mistakes to Avoid

Treating $f(x - h)$ as a shift left is wrong because changes inside the input work oppositely; $f(x - h)$ shifts the graph right $h$ units when $h > 0$ .
Forgetting the reciprocal in horizontal scaling is wrong because $g(x) = f(bx)$ changes widths by $\frac{1}{|b|}$ , not by $|b|$ .
Confusing $-f(x)$ and $f(-x)$ is wrong because $-f(x)$ reflects across the $x$ -axis, while $f(-x)$ reflects across the $y$ -axis.
Applying transformations in a random order can give the wrong equation or graph; use $g(x) = a f(b(x - h)) + k$ to identify horizontal changes, then vertical changes.
Ignoring parentheses in expressions like $f(2(x - 3))$ is wrong because the horizontal shift is $3$ units right, not $6$ units right.

Practice Questions

1 Describe all transformations from $f(x) = x^2$ to $g(x) = -2(x - 3)^2 + 5$ .
2 If the point $(4, 7)$ is on $f(x)$ , find the corresponding point on $g(x) = 3f(2(x - 1)) - 4$ .
3 Write a transformed function based on $f(x) = \sqrt{x}$ that shifts left $2$ units, reflects across the $x$ -axis, and shifts up $6$ units.
4 Explain why $g(x) = f(x + 4)$ moves the graph left instead of right, even though $4$ is positive.

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Pre-Calculus Function Transformations Cheat Sheet

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