Translations of Functions- Transformations Explained

What Function Transformations Actually Are

Function transformations are how we move, flip, and resize graphs. That's it. You take a basic function and change its equation to get a new graph position or shape.

Every transformation follows predictable rules. Once you know the rules, you can look at any transformed function and know exactly what its graph looks like—no guessing needed.

The Four Basic Types of Transformations

All function transformations fall into four categories:

• Translations — sliding the graph up, down, left, or right
• Reflections — flipping the graph over an axis
• Stretches — pulling the graph taller or wider
• Compressions — squishing the graph shorter or narrower

Most functions you'll see combine these. A single equation might have a vertical shift, a reflection, and a horizontal stretch all at once.

Translations: Moving the Graph

Vertical Translations

Adding or subtracting a number outside the function moves the graph up or down.

For y = f(x) + k:

• If k is positive, the graph shifts up by k units
• If k is negative, the graph shifts down by |k| units

Example: y = x² + 3 is just the basic parabola moved up 3 units. y = x² - 2 moves it down 2 units.

The shape stays identical. Only the position changes.

Horizontal Translations

Adding or subtracting a number inside the function moves the graph left or right. But here's where students mess up—it's backwards from what you'd expect.

For y = f(x - h):

• If h is positive, the graph shifts right by h units
• If h is negative, the graph shifts left by |h| units

Example: y = (x - 4)² looks like x² shifted right 4 units. y = (x + 2)² shifts left 2 units—yes, even though it says +2, the graph goes left.

The sign inside the parentheses is opposite of the direction. Remember this or you'll get every horizontal translation wrong.

Reflections: Flipping the Graph

Reflections flip the graph over a line. Two common ones:

Reflection Over the X-Axis

y = -f(x) flips the graph vertically, over the x-axis.

Every y-value becomes its opposite. Points above the axis go below it, and vice versa. The x-axis acts like a mirror.

Reflection Over the Y-Axis

y = f(-x) flips the graph horizontally, over the y-axis.

Points on the right side of the y-axis jump to the left, and vice versa. The y-axis is the mirror line.

These are easy to spot. Negative sign outside the function? Flip vertically. Negative sign inside the function? Flip horizontally.

Stretches and Compressions

Vertical Stretches and Compressions

Multiplying the outside of the function by a number stretches or compresses vertically.

For y = a · f(x):

• If |a| > 1, the graph stretches vertically—it's taller
• If 0 < |a| < 1, the graph compresses vertically—it's shorter
• If a is negative, you also get a reflection over the x-axis

Example: y = 2x² stretches the basic parabola to be twice as tall. y = 0.5x² squishes it to half the height.

Horizontal Stretches and Compressions

Multiplying the inside of the function by a number stretches or compresses horizontally.

For y = f(bx):

• If |b| > 1, the graph compresses horizontally—it's narrower
• If 0 < |b| < 1, the graph stretches horizontally—it's wider
• If b is negative, you also get a reflection over the y-axis

Notice: inside multiplication works backwards. Bigger number inside means narrower graph, not wider. This confuses people constantly.

Reading Transformations from an Equation

Most functions combine multiple transformations. To decode them, break down the equation piece by piece.

General form: y = a · f(b(x - h)) + k

• a — vertical stretch/compression (and reflection if negative)
• b — horizontal stretch/compression (and reflection if negative)
• h — horizontal shift (opposite sign of what's in parentheses)
• k — vertical shift (same sign as what's written)

Work from the inside out: start with the horizontal transformations (h and b), then do the vertical ones (a and k).

Quick Reference: Transformation Rules

How to Graph Transformed Functions

Step 1: Identify the parent function—the basic function before any transformations.

Step 2: List all transformations in order: horizontal shifts, horizontal stretches/compressions, reflections, vertical stretches/compressions, vertical shifts.

Step 3: Apply them one at a time to key points from the parent function.

Step 4: Plot the transformed points and connect them in the same pattern as the parent.

Pick 3-5 anchor points from the original graph—usually intercepts and vertices. Transform those points, then sketch the rest.

Example: Graphing y = -2(x - 1)² + 3

Starting with y = x² (parent function):

• (x - 1) — shift right 1 unit
• The 2 outside — vertical stretch by factor of 2
• Negative sign — reflect over x-axis
• +3 at end — shift up 3 units

Take a point from x², like (0, 0). Apply each transformation:

• (0, 0) → shift right 1 → (1, 0)
• Stretch by 2 → (1, 0)
• Reflect over x-axis → (1, 0)
• Shift up 3 → (1, 3)

Repeat for other anchor points, then sketch the parabola. You get an upside-down parabola, taller than the original, with its vertex at (1, 3).

What to Watch Out For

Horizontal transformations look backwards. Inside the function, larger numbers mean the graph gets closer to the y-axis, not farther. This trips up almost everyone.

Order matters. When multiple transformations hit the same point, the result depends on which you apply first. Stick to the standard order: inside transformations first, then outside.

Stretching and shifting don't commute. If you stretch first, then shift, you get a different result than shifting then stretching. The equation order tells you which happens in which sequence.