1.5 Transformation of Functions

Function Transformations

Function transformations let you take a parent function you already know and modify its graph by shifting, stretching, compressing, or reflecting it. Instead of graphing every new function from scratch, you can start with a familiar shape and apply predictable changes to get the new one.

This is one of the most practical skills in pre-calc. Once you understand how each transformation works, you can quickly sketch graphs, interpret equations, and build complex functions out of simple ones.

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Vertical and Horizontal Shifts

Shifts (also called translations) slide the entire graph without changing its shape or size.

Vertical shifts add a constant $k$ to the output: $f(x) + k$

• Positive $k$ shifts the graph up (e.g., $f(x) + 3$ moves every point up 3 units)
• Negative $k$ shifts the graph down (e.g., $f(x) - 2$ moves every point down 2 units)

Horizontal shifts add a constant $h$ inside the function argument: $f(x - h)$

• Positive $h$ shifts the graph right (e.g., $f(x - 1)$ shifts right 1 unit)
• Negative $h$ shifts the graph left (e.g., $f(x + 4) = f(x - (-4))$ shifts left 4 units)

The horizontal direction is counterintuitive for most people. The key: the shift goes in the opposite direction of the sign you see inside the parentheses. If you see $f(x + 4)$, the graph moves left, not right. Think of it this way: the function "reaches" its original output values sooner (4 units to the left).

Reflections Across Axes

Reflections flip the graph over an axis.

• Reflection across the x-axis: Multiply the entire function by $-1$ to get $-f(x)$. This negates every y-value, flipping the graph vertically.
  • Example: if $f(x) = x^2$, then $-f(x) = -x^2$ turns the upward-opening parabola into a downward-opening one.
• Reflection across the y-axis: Replace $x$ with $-x$ to get $f(-x)$. This mirrors the graph horizontally.
  • Example: if $f(x) = x^3$, then $f(-x) = (-x)^3 = -x^3$ reflects the cubic across the y-axis.

A quick way to remember: negating outside the function ($-f(x)$) flips over the x-axis. Negating inside the function ($f(-x)$) flips over the y-axis.

Even and Odd Functions

Symmetry and reflections connect directly to the idea of even and odd functions.

• Even functions satisfy $f(-x) = f(x)$ for all $x$ in the domain. Their graphs are symmetric about the y-axis, meaning reflecting across the y-axis produces the same graph.
  • Example: $f(x) = x^2$ is even because $f(-x) = (-x)^2 = x^2 = f(x)$
• Odd functions satisfy $f(-x) = -f(x)$ for all $x$ in the domain. Their graphs are symmetric about the origin, meaning a 180° rotation around the origin produces the same graph.
  • Example: $f(x) = x^3$ is odd because $f(-x) = (-x)^3 = -x^3 = -f(x)$
• Neither even nor odd: Most functions don't have either type of symmetry.
  • Example: $f(x) = x^2 + x$ is neither, because $f(-x) = x^2 - x$, which equals neither $f(x)$ nor $-f(x)$

To test algebraically, compute $f(-x)$ and compare it to both $f(x)$ and $-f(x)$. If it matches $f(x)$, the function is even. If it matches $-f(x)$, it's odd. If neither, it's neither.

Compressions and Stretches

Scaling transformations change how "wide" or "tall" the graph is.

Vertical scaling multiplies the function by a constant $a$: $a \cdot f(x)$

• $|a| > 1$ stretches the graph vertically (e.g., $2f(x)$ doubles every y-value, making the graph taller)
• $0 < |a| < 1$ compresses the graph vertically (e.g., $\frac{1}{3}f(x)$ shrinks every y-value to one-third, making the graph flatter)
• If $a$ is negative, you also get a reflection across the x-axis combined with the scaling

Horizontal scaling multiplies $x$ by a constant $b$ inside the function: $f(bx)$

• $|b| > 1$ compresses the graph horizontally (e.g., $f(2x)$ squeezes the graph to half its original width)
• $0 < |b| < 1$ stretches the graph horizontally (e.g., $f(\frac{1}{4}x)$ stretches the graph to 4 times its original width)
• If $b$ is negative, you also get a reflection across the y-axis combined with the scaling

Just like with shifts, horizontal scaling behaves opposite to what you might expect. A larger value of $b$ inside $f(bx)$ makes the graph narrower, not wider. The x-values need to be smaller to produce the same outputs, so everything gets compressed toward the y-axis.

Combining Function Transformations

When multiple transformations are applied at once, order matters. Follow this sequence:

1. Horizontal scaling and reflection (the $b$ factor inside the function)
2. Horizontal shift (the $h$ value inside the function)
3. Vertical scaling and reflection (the $a$ factor outside the function)
4. Vertical shift (the $k$ value added outside)

The general form of a fully transformed function is:

$a \cdot f\left(\frac{x - h}{b}\right) + k$

where:

• $a$ = vertical scale/reflection factor
• $b$ = horizontal scale/reflection factor
• $h$ = horizontal shift (positive shifts right)
• $k$ = vertical shift (positive shifts up)

Example: Given $f(x) = x^2$, describe the transformations in $g(x) = -2f(3x + 1) - 4$.

First, rewrite the inside to match the general form. Factor out the horizontal scale from the argument:

$3x + 1 = 3\left(x + \frac{1}{3}\right) = 3\left(x - \left(-\frac{1}{3}\right)\right)$

Now you can read off the parameters:

• $a = -2$: vertical stretch by a factor of 2, reflected across the x-axis
• $b = \frac{1}{3}$: horizontal compression by a factor of 3
• $h = -\frac{1}{3}$: horizontal shift left by $\frac{1}{3}$ units
• $k = -4$: vertical shift down 4 units

Advanced Function Concepts

Several related ideas build on transformations and come up throughout the rest of the course:

• Function composition combines two functions by using the output of one as the input of another, written $(f \circ g)(x) = f(g(x))$
• Inverse functions reverse a function's input and output. If $f(a) = b$, then $f^{-1}(b) = a$
• Piecewise functions use different rules for different intervals of the domain
• One-to-one functions produce a unique output for every input (they pass the horizontal line test), which is the requirement for an inverse function to exist