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Transformations of
Functions & Graphs

Master how replacing \( f(x) \) by \( f(x)+k \), \( kf(x) \), \( f(kx) \), and \( f(x+k) \) shifts, stretches, and reflects every graph you'll ever meet.

Algebra 2

What Is a Transformation?

A transformation is a rule that takes every point on the graph of a parent function \( f(x) \) and moves it to a new location, producing a new graph \( g(x) \). The shape may shift, stretch, compress, or flip—but it always starts from a parent function you already know.

The Four Families

\( f(x) + k \)
Vertical shift: moves the graph up or down
\( f(x + k) \)
Horizontal shift: moves the graph left or right
\( k \cdot f(x) \)
Vertical stretch/compress and reflection over \( x \)-axis
\( f(kx) \)
Horizontal stretch/compress and reflection over \( y \)-axis

Why Transformations Matter

Every function you study in Algebra 2—quadratics, absolute value, square root, exponentials, and logarithms—can be graphed by transforming its parent. Instead of plotting dozens of points, you sketch the parent and apply transformations. This skill also lets you read a graph backwards: given a shifted curve, you can recover the equation.

Vertical Shifts: \( f(x) + k \)

x y
\(f(x) = x^2\)
\(f(x) + 3\)
\(f(x) - 2\)

Horizontal Shifts: \( f(x + k) \)

x y
\(f(x + 3)\)
\(f(x - 2)\)
\(f(x)\)

Vertical Stretch: \( k \cdot f(x) \)

x y
\(f(x)\)
\(2f(x)\)
\(0.5f(x)\)

Horizontal Stretch: \( f(kx) \)

x y
\(f(x)\)
\(f(2x)\)
\(f(0.5x)\)

Inside vs. Outside the Function

Inside changes
\( f(\textcolor{#818cf8}{k}x + \textcolor{#818cf8}{h}) \): affects the input. These produce horizontal transformations and act in the opposite direction you'd expect.
Outside changes
\( \textcolor{#fbbf24}{a} \cdot f(x) + \textcolor{#fbbf24}{k} \): affects the output. These produce vertical transformations and act in the expected direction.

Reflections: Flipping the Graph

\( -f(x) \)
Reflect over the \( x \)-axis. Every \( y \)-value becomes its opposite: \( (x, y) \to (x, -y) \).
\( f(-x) \)
Reflect over the \( y \)-axis. Every \( x \)-value becomes its opposite: \( (x, y) \to (-x, y) \).

Even and Odd Functions

Even: \(f(-x) = f(x)\)
\(x^2\)
Odd: \(f(-x) = -f(x)\)
\(x^3\)

How to Apply Transformations

1
Identify the parent function. Recognize \( x^2 \), \( |x| \), \( \sqrt{x} \), \( 2^x \), etc.
2
Rewrite in standard form: \( g(x) = a \cdot f(b(x - h)) + k \).
3
Apply horizontal transformations first (inside the function): stretch/compress by \( \frac{1}{b} \), then shift by \( h \).
4
Apply vertical transformations next (outside the function): stretch/compress by \( |a| \), reflect if \( a < 0 \), then shift by \( k \).
5
Plot key reference points from the parent through each transformation step.

Example 1: \( g(x) = (x - 3)^2 + 2 \)

0 1 2 3 4 5 6
\(f(x) = x^2\)
 (3, 2)
\(g(x) = (x-3)^2 + 2\)

Example 2: \( g(x) = -2|x + 1| + 4 \)

0 -1 1 2 -2
\(|x|\)
 (-1, 4)
\(g(x)\)

Example 3: \( g(x) = \sqrt{-(x - 4)} \)

x y 0 1 2 3 4 -1 -2
\(\sqrt{x}\)
 (4, 0)
\(g(x)\)

Common Mistakes to Avoid

✕ Wrong horizontal direction
Thinking \( f(x + 3) \) shifts right. It shifts left. The sign inside is opposite to the direction.
✕ Confusing stretch with shift
Treating \( 2f(x) \) as a shift up by 2. It is a vertical stretch by factor 2, not a shift.
✕ Wrong order of operations
Applying vertical shift before stretch. Always apply stretches/reflections first, then shifts.

Summary Table

Transformation Effect Direction
\( f(x) + k \) Vertical shift Up if \( k > 0 \)
\( f(x + k) \) Horizontal shift Left if \( k > 0 \)
\( k \cdot f(x) \) Vertical stretch/compress Stretch if \( |k| > 1 \)
\( f(kx) \) Horizontal stretch/compress Compress if \( |k| > 1 \)
\( -f(x) \) Reflection Over \( x \)-axis
\( f(-x) \) Reflection Over \( y \)-axis

Question 1 of 5

The graph of \( g(x) = (x + 5)^2 - 3 \) is obtained from \( f(x) = x^2 \) by shifting:

Question 2 of 5

Which transformation does \( g(x) = 3f(x) \) perform on the graph of \( f(x) \)?

Question 3 of 5

If \( f(x) = x^3 \), what is the effect of replacing \( f(x) \) with \( f(-x) \)?

Question 4 of 5

The function \( h(x) = f(2x) \) is graphed. Compared to \( f(x) \), the graph of \( h(x) \) is:

Question 5 of 5

Which of the following is an even function?