Function Transformations

Why This Matters

Function transformations are the universal language for describing how graphs change—and they show up everywhere in algebra and trigonometry. Whether you're graphing a shifted parabola, analyzing a stretched sine wave, or modeling real-world phenomena, you're applying the same core principles. Exams test whether you can predict how a graph will look based on its equation, and more importantly, whether you understand why each transformation behaves the way it does.

Here's the key insight: every transformation either affects the input (what happens to $x$ before the function processes it) or the output (what happens to $y$ after the function produces it). This input-output framework explains why horizontal transformations seem to work "backwards" while vertical ones feel intuitive. Don't just memorize that $f(x + 2)$ shifts left—understand that you're feeding the function values 2 units earlier, so the graph responds 2 units sooner. Master this logic, and you'll handle any transformation problem with confidence.

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Shifts: Moving the Graph Without Changing Its Shape

Shifts (or translations) relocate a graph while preserving its exact shape and size. The function's output values stay the same—they just occur at different locations on the coordinate plane.

Vertical Shifts $f(x) + k$

• Adding $k$ to the output shifts the entire graph up or down—every point moves the same vertical distance
• Positive $k$ shifts up; negative $k$ shifts down—this follows your intuition since you're directly adding to $y$-values
• Key points transform predictably: if $(2, 5)$ is on $f(x)$, then $(2, 5+k)$ is on $f(x) + k$

Horizontal Shifts $f(x + h)$

• Adding $h$ inside the function shifts the graph horizontally—but in the opposite direction you might expect
• Positive $h$ shifts left; negative $h$ shifts right—think of it as the function "reaching" its values earlier or later
• The counterintuitive direction occurs because $f(x + 2) = 3$ when $x = -2$, not when $x = 2$

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Stretches and Compressions: Scaling the Graph

These transformations change the graph's dimensions by multiplying coordinates by a scale factor. Stretches pull the graph away from an axis; compressions push it toward that axis.

Vertical Stretches and Compressions $af(x)$

• Multiplying the output by $a$ scales all $y$-values—the graph stretches away from or compresses toward the $x$-axis
• $|a| > 1$ stretches vertically; $0 < |a| < 1$ compresses—a factor of 2 doubles heights, a factor of $\frac{1}{2}$ halves them
• $x$-intercepts stay fixed since $a \cdot 0 = 0$—these are your anchor points when graphing

Horizontal Stretches and Compressions $f(bx)$

• Multiplying the input by $b$ scales all $x$-values—but the effect is inverse to what you might expect
• $|b| > 1$ compresses horizontally; $0 < |b| < 1$ stretches—$f(2x)$ makes the graph narrower, not wider
• $y$-intercept stays fixed since $f(b \cdot 0) = f(0)$—this point anchors horizontal scaling

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Reflections: Flipping the Graph Across an Axis

Reflections create mirror images of the original graph across a line. Each point maps to its mirror counterpart at equal distance from the axis of reflection.

Reflection Over the $x$-Axis $-f(x)$

• Negating the output flips the graph vertically—points above the $x$-axis move below, and vice versa
• $x$-intercepts remain unchanged since negating zero still gives zero—these points lie on the mirror line
• Peaks become valleys and valleys become peaks—maximum values become minimum values

Reflection Over the $y$-Axis $f(-x)$

• Negating the input flips the graph horizontally—the left and right sides swap positions
• $y$-intercept remains unchanged since $f(-0) = f(0)$—this point lies on the mirror line
• Tests for even functions: if $f(-x) = f(x)$, the function is symmetric about the $y$-axis

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Special Transformations: Absolute Value and Composites

These transformations build on the basics but introduce new graphical behaviors. They're especially useful for creating piecewise-looking graphs from smooth functions.

Absolute Value Transformation $|f(x)|$

• Taking the absolute value of the output reflects negative portions above the $x$-axis—any part of the graph below the axis "bounces" up
• Positive portions remain unchanged—only negative $y$-values are affected
• Creates sharp corners at $x$-intercepts where the original graph crossed the axis—these corners indicate non-differentiability

Composite Transformations

• Multiple transformations applied in sequence follow a specific order: horizontal shifts/scales first (inside), then vertical shifts/scales (outside)
• Order matters critically—$2f(x) + 3$ (stretch then shift) differs from $2(f(x) + 3) = 2f(x) + 6$
• Standard form $a \cdot f(b(x - h)) + k$ reveals all transformations: $a$ = vertical scale, $b$ = horizontal scale, $h$ = horizontal shift, $k$ = vertical shift

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Parent Functions: Your Starting Points

Understanding parent functions gives you a reference point for all transformations. Every transformed function is just a parent function that's been shifted, scaled, or reflected.

Recognizing and Using Parent Functions

• Common parent functions include $f(x) = x$, $x^2$, $x^3$, $\sqrt{x}$, $|x|$, $\frac{1}{x}$, and trigonometric functions
• Key features to memorize: intercepts, domain/range, symmetry, and general shape for each parent
• Transformation problems typically start here—identify the parent function first, then apply transformations step-by-step using key points

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Quick Reference Table

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Self-Check Questions

1. If the point $(3, -4)$ is on $f(x)$, what point is on $-2f(x - 1) + 5$?

2. Which two transformations both preserve the $y$-intercept of a function, and why?

3. Compare and contrast the graphs of $|f(x)|$ and $f(|x|)$ for a function like $f(x) = x - 2$. How do they differ?

4. A student claims that $f(x + 3)$ shifts the graph right because "you're adding 3." Explain the error in their reasoning.

5. Given $g(x) = -\frac{1}{2}(x + 4)^2 - 3$, identify the parent function and describe each transformation in the order they should be applied when graphing.