What are transformations of linear functions?

A transformation is a change made to the graph of a function.
For linear functions (lines), transformations change position, steepness, or direction, but the graph stays a straight line.

The basic form of a linear function is:

Where:

• m = slope (steepness)

• b = y-intercept (starting point on y-axis)

Types of transformations

1️⃣ Vertical shifts (up or down)

• Add or subtract a number outside the function.

• Example:

Effect:

• The line moves up 3 units.

• Slope stays the same.

2️⃣ Horizontal shifts (left or right)

• Add or subtract a number inside the function (with x).

• Example:

Effect:

• The line moves right 4 units.

• Slope stays the same.

Tip: (x − h) → move right h
(x + h) → move left h

3️⃣ Vertical stretching/shrinking

• Multiply the function by a number.

• Example:

Effect:

• Line becomes steeper.

• Larger numbers → steeper slope

• Smaller numbers (between 0 and 1) → flatter slope

4️⃣ Reflection (flipping)

• Multiply by −1.

• Example:

Effect:

• Line flips over the x-axis.

• Positive slope becomes negative, negative becomes positive.

Combining transformations

You can combine shifts, stretches, and reflections in one function.

Example:

Step-by-step:

1. (x − 3) → move right 3

2. −2 → slope is −2 (flip and steepen)

3. +4 → move up 4

Visual summary

Common beginner mistakes

• ❌ Confusing inside vs. outside the function
  ✅ Inside (with x) → horizontal, Outside → vertical

• ❌ Forgetting slope changes with multiplication
  ✅ Always check slope when stretching or reflecting

• ❌ Thinking shifts change slope
  ✅ Only slope multipliers or reflections change steepness

Why transformations matter

• Makes graphing linear functions fast and easy

• Helps with real-world problems like speed, cost, or growth

• Sets the foundation for quadratics and higher functions