1.12 Transformations of Functions

What are transformations of functions in AP Precalculus?

Transformations let you build a new function from a known one by shifting, stretching, shrinking, or reflecting its graph. Additive changes (adding or subtracting a constant) move the graph, while multiplicative changes (multiplying by a constant) stretch, compress, or flip it. Once you can read the form of an equation like $g(x) = af(b(x+h)) + k$, you can predict the new graph and how the domain and range change.

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Why This Matters for the AP Precalculus Exam

This topic shows up across all four units because every function family (polynomial, rational, exponential, logarithmic, trigonometric) can be transformed the same way. On the AP Precalculus exam, you may need to:

• Recognize a transformation from an equation, table, or graph.
• Construct a new function rule when given a parent function and a described shift, dilation, or reflection.
• Explain how the domain and range change after a transformation.
• Connect the structure of an equation to the behavior of its graph.

Free-response questions in this course ask you to justify conclusions, not just state them. Being able to clearly describe what each constant does to a graph helps you write precise explanations and keeps your work easy to follow.

Key Takeaways

• $g(x) = f(x) + k$ shifts the graph up by $k$ units (down if $k$ is negative).
• $g(x) = f(x + h)$ shifts the graph by $-h$ units horizontally: $+h$ moves it left, $-h$ moves it right.
• $g(x) = af(x)$ is a vertical dilation by a factor of $|a|$; if $a < 0$, it also reflects over the x-axis.
• $g(x) = f(bx)$ is a horizontal dilation by a factor of $\left|\frac{1}{b}\right|$; if $b < 0$, it also reflects over the y-axis.
• Translations move the graph without changing its shape; dilations and reflections change size or orientation.
• A transformation can change the domain and range compared to the parent function.

Additive Transformations (Translations)

An additive transformation of a function involves adding or subtracting a constant value, either to the output or inside the input.

Vertical Translations

The function $g(x) = f(x) + k$ is an additive transformation of $f$. The constant $k$ produces a vertical translation of the graph by $k$ units. If $k > 0$ the graph moves up; if $k < 0$ the graph moves down.

The shape stays the same. Each output value is just raised or lowered by $k$, so the point $(x, y)$ moves to $(x, y + k)$.

Horizontal Translations

The function $g(x) = f(x + h)$ is also an additive transformation, but the constant is added inside the function. This produces a horizontal translation of the graph by $-h$ units. The sign flips your intuition: $f(x + 3)$ moves the graph 3 units left, and $f(x - 3)$ moves it 3 units right.

In point terms, $(x, y)$ moves to $(x - h, y)$.

For both vertical and horizontal translations, the shape of the graph does not change. Only its position on the coordinate plane changes. If you know the graph of $f$, you can sketch $g$ by applying the correct shifts.

Multiplicative Transformations (Dilations and Reflections)

A multiplicative transformation involves multiplying by a constant, either the whole output or the input.

Vertical Dilations

The function $g(x) = af(x)$, where $a \neq 0$, is a multiplicative transformation that scales the graph vertically by a factor of $|a|$. The distance from each point to the x-axis is multiplied by $|a|$.

• If $a > 1$, the graph is stretched vertically and looks "taller."
• If $0 < a < 1$, the graph is compressed vertically and looks "shorter."
• If $a < 0$, the transformation also includes a reflection over the x-axis, flipping the graph.

In point terms, $(x, y)$ moves to $(x, a \cdot y)$.

Horizontal Dilations

The function $g(x) = f(bx)$, where $b \neq 0$, scales the graph horizontally by a factor of $\left|\frac{1}{b}\right|$. This one is counterintuitive because the factor uses the reciprocal of $b$.

• If $|b| > 1$, the graph is compressed horizontally and looks "narrower."
• If $0 < |b| < 1$, the graph is stretched horizontally and looks "wider."
• If $b < 0$, the transformation also includes a reflection over the y-axis, flipping the graph.

In point terms, $(x, y)$ moves to $\left(\frac{x}{b}, y\right)$.

For both vertical and horizontal dilations, the basic shape stays similar but the size changes. Reflections flip the orientation. If you know the graph of $f$, you can sketch $g$ by applying the correct dilation and any reflection.

Working with Combined Transformations

Additive and multiplicative transformations can be combined into a single function. Consider:

$g(x) = a \cdot f(b(x + h)) + k$

Here you have four constants doing four jobs:

• $b$ produces a horizontal dilation by a factor of $\left|\frac{1}{b}\right|$ (and a y-axis reflection if $b < 0$).
• $h$ produces a horizontal translation by $-h$ units.
• $a$ produces a vertical dilation by a factor of $|a|$ (and an x-axis reflection if $a < 0$).
• $k$ produces a vertical translation by $k$ units.

A useful pattern to remember: anything inside the function (with the input $x$) affects the graph horizontally and often behaves opposite to what the sign suggests. Anything outside the function affects the graph vertically and behaves the way you expect.

A transformation can also change the domain and range of a function compared to its parent function. Horizontal translations and dilations can shift or scale the domain, while vertical translations and dilations can shift or scale the range. Always check whether features like asymptotes, endpoints, or restricted values move when you transform a function.

How to Use This on the AP Precalculus Exam

MCQ

• Match an equation to its graph by checking each constant one at a time: vertical shift ($k$), horizontal shift ($h$), vertical dilation/reflection ($a$), horizontal dilation/reflection ($b$).
• Watch the signs inside the function. $f(x + 2)$ moves left, not right.
• When a problem gives you a table, track how specific points move. Use the mappings $(x, y) \to (x - h, y + k)$ and $(x, y) \to \left(\frac{x}{b}, a \cdot y\right)$.

Free Response

• When asked to construct a transformed function, write the new rule explicitly and state what each constant does.
• When explaining a graph, name the transformation (translation, dilation, or reflection), the direction, and the size of the change.
• If the question involves domain or range, state how each changes and why, referencing the specific shift or dilation that caused it.

Common Trap

• Mixing up horizontal stretch and compression. Remember $|b| > 1$ compresses horizontally and $0 < |b| < 1$ stretches horizontally.
• Forgetting that the horizontal factor is $\left|\frac{1}{b}\right|$, not $|b|$.

Common Misconceptions

• "$f(x + h)$ shifts right because of the plus sign." Inside changes work in reverse. $f(x + h)$ shifts the graph $-h$ units, so a plus moves it left.
• "$f(bx)$ stretches the graph by a factor of $b$." The horizontal scale factor is $\left|\frac{1}{b}\right|$. A larger $|b|$ compresses the graph; a smaller $|b|$ stretches it.
• "A reflection always changes the domain." A reflection over the x-axis ($a < 0$) flips outputs and changes the range, not the domain. A reflection over the y-axis ($b < 0$) flips inputs. Whether the domain or range actually changes depends on the parent function and any restrictions, so check each case instead of assuming.
• "Vertical and horizontal transformations behave the same way." Outside changes (vertical) match the sign you expect; inside changes (horizontal) often work opposite to the sign.
• "Translations change the shape of the graph." Translations only move the graph. Its shape stays identical; only dilations and reflections change size or orientation.

Related AP Precalculus Guides

• 1.8 Rational Functions and Zeros
• 1.4 Polynomial Functions and Rates of Change
• 1.7 Rational Functions and End Behavior
• 1.11 Equivalent Representations of Polynomial and Rational Expressions
• 1.3 Rates of Change in Linear and Quadratic Functions
• 1.9 Rational Functions and Vertical Asymptotes