Graph transformations

Welcome to Graph Transformations!

Ever wondered how computer animators move characters across a screen or how audio engineers adjust sound waves? They use graph transformations! In this chapter, we’ll learn how to take a basic "parent" function—like \(y = x^2\)—and shift, stretch, or flip it to create any graph we want. Don’t worry if it seems like a lot of rules at first; once you see the patterns, it becomes as simple as using a photo editing app.

1. The Golden Rule: Inside vs. Outside

Before we dive into specific moves, there is one "Golden Rule" that will make this entire chapter easier. It’s all about where the number is added or multiplied in the equation \(y = f(x)\).

• Outside the brackets: These changes affect the y-coordinates. They are "honest" and do exactly what you’d expect (e.g., \(+3\) moves the graph up).
• Inside the brackets: These changes affect the x-coordinates. They are "counter-intuitive" and do the opposite of what you’d expect (e.g., \(+3\) moves the graph left).

Quick Review Box:
Outside = Vertical (y) = Follows logic.
Inside = Horizontal (x) = Does the opposite.

2. Translations (Shifting the Graph)

A translation slides the graph up, down, left, or right without changing its shape or orientation.

Vertical Translation: \(y = f(x) + a\)

Adding a constant outside the function shifts it vertically.
- \(y = f(x) + a\): Moves up by \(a\) units.
- \(y = f(x) - a\): Moves down by \(a\) units.

Horizontal Translation: \(y = f(x + a)\)

Adding a constant inside the brackets shifts it horizontally. Remember: Inside is the opposite!
- \(y = f(x + a)\): Moves left by \(a\) units.
- \(y = f(x - a)\): Moves right by \(a\) units.

Vector Notation:
In your exam, you may be asked to describe these using a column vector \(\begin{pmatrix} x \\ y \end{pmatrix}\).
A translation of \(y = f(x - 3) + 2\) would be described by the vector \(\begin{pmatrix} 3 \\ 2 \end{pmatrix}\).

Key Takeaway: To move a graph, add or subtract. Outside affects height (y); inside affects side-to-side (x) but switches the sign.

3. Stretches (Resizing the Graph)

Stretches pull the graph away from an axis or squash it toward an axis. We call the multiplier the scale factor.

Vertical Stretch: \(y = a \cdot f(x)\)

Multiplying outside the function stretches it vertically.
- This is a stretch of scale factor \(a\) parallel to the y-axis.
- All y-coordinates are multiplied by \(a\). The x-coordinates stay the same.

Horizontal Stretch: \(y = f(ax)\)

Multiplying inside the brackets stretches it horizontally. Remember: Inside is the opposite!
- This is a stretch of scale factor \(\frac{1}{a}\) parallel to the x-axis.
- All x-coordinates are divided by \(a\). The y-coordinates stay the same.

Did you know?
If you have \(y = f(2x)\), you aren't making the graph twice as wide; you are actually squashing it to be half as wide! Think of it like a spring: as the number inside gets bigger, the "tension" increases and the graph compresses.

Summary Table for Stretches:
- \(y = 3f(x)\): Scale factor 3, parallel to y-axis (Vertical).
- \(y = f(3x)\): Scale factor \(\frac{1}{3}\), parallel to x-axis (Horizontal).

4. Reflections (Flipping the Graph)

Reflections are just special stretches where the multiplier is \(-1\).

Reflection in the x-axis: \(y = -f(x)\)

The minus is outside. It affects the y-values, turning positive heights into negative depths. This flips the graph upside down over the x-axis.

Reflection in the y-axis: \(y = f(-x)\)

The minus is inside. It affects the x-values, turning "right" into "left." This flips the graph sideways over the y-axis.

Memory Aid: "In-Y-Out-X"
Minus Inside = Reflection in Y-axis.
Minus Outside = Reflection in X-axis.

5. Combining Transformations

When you have more than one transformation, the order usually matters! For OCR H240, you need to be able to handle combinations like \(y = a \cdot f(x + b)\).

The "Safe" Order:
Generally, follow the order of operations (BIDMAS).
1. Handle the inside (horizontal moves) first.
2. Handle the outside (vertical moves) second.

Example: \(y = 2f(x - 5)\)
Step 1: Translate by the vector \(\begin{pmatrix} 5 \\ 0 \end{pmatrix}\) (Move right 5).
Step 2: Stretch with scale factor 2 parallel to the y-axis (Double the height).

Common Mistake to Avoid:
Don't mix up the scale factor for horizontal stretches. If you see \(f(4x)\), the scale factor is \(\frac{1}{4}\), not 4!

6. The Modulus of Linear Functions: \(y = |ax + b|\)

The modulus sign \(| \dots |\) means "take the absolute value" (ignore the minus sign). In Stage 2, you need to sketch the modulus of linear functions like \(y = |2x - 3|\).

Step-by-Step Sketching:
1. Sketch the line \(y = ax + b\) as if the modulus wasn't there (using a dotted line for the parts below the x-axis).
2. Identify the "negative" part: Any part of the line that is below the x-axis (where \(y < 0\)).
3. Reflect it: Flip those negative parts upwards over the x-axis so they become positive.
4. The result usually looks like a "V" shape.

Analogy: Think of the x-axis as a floor. The modulus function is like a bouncy ball; it can’t go through the floor. As soon as it hits the floor, it bounces back up!

Key Takeaway: For \(y = |f(x)|\), nothing is allowed to be negative. Flip all the "underground" parts to be "above ground."

Final Checklist for Success

• Can I distinguish between \(f(x) + a\) and \(f(x + a)\)?
• Do I remember that horizontal changes are the reciprocal (for stretches) or the opposite sign (for translations)?
• Can I use column vectors for translations?
• Can I sketch a "V" shaped modulus graph?

Don't worry if this seems tricky at first! Try sketching \(y = x^2\) and then \(y = (x-2)^2 + 3\) on the same axes to see the rules in action. Practice is the best way to make these "edits" feel natural.