Introduction: Playing with Graphs
Welcome! In this chapter, we are going to learn how to take a "parent" graph (the original shape) and move it, stretch it, or squash it. Think of Graph Transformations like photo filters or resizing a window on your computer—the basic picture stays the same, but its position and size change.
Mastering this is important because it allows you to sketch complex-looking equations very quickly without having to calculate dozens of coordinates. Let's dive in!
1. The Golden Rule: Inside vs. Outside
Before we look at specific moves, there is one "trick" that will help you throughout this entire chapter. We look at where the change is happening in the equation \( y = f(x) \).
The "Outside" Changes: If the change is outside the brackets (affecting the whole function), it affects the vertical direction (\(y\)-axis). These are "honest" changes—they do exactly what they say!
The "Inside" Changes: If the change is inside the brackets (next to the \(x\)), it affects the horizontal direction (\(x\)-axis). These are "rebellious" changes—they usually do the opposite of what you'd expect!
Quick Review:
- Outside the bracket = Vertical change (\(y\)). Honest/Logical.
- Inside the bracket = Horizontal change (\(x\)). Rebellious/Opposite.
2. Translations (Shifting the Graph)
A Translation simply slides the graph up, down, left, or right without changing its shape or orientation.
Vertical Translation: \( y = f(x) + a \)
This moves the graph up or down. Because the \( a \) is outside the brackets, it is a vertical move and it is "honest."
- \( y = f(x) + 3 \): Move the graph up by 3 units.
- \( y = f(x) - 5 \): Move the graph down by 5 units.
Horizontal Translation: \( y = f(x + a) \)
This moves the graph left or right. Because the \( a \) is inside the brackets, it is a horizontal move and it does the opposite of the sign.
- \( y = f(x + 2) \): You might think "plus means right," but it actually moves Left 2 units.
- \( y = f(x - 4) \): Moves Right 4 units.
Using Column Vectors
In your exam, you might be asked to describe a translation using a vector: \(\begin{pmatrix} x \\ y \end{pmatrix}\).
The top number is the horizontal move, and the bottom number is the vertical move.
Example: To describe the transformation from \( y = f(x) \) to \( y = f(x - 3) + 2 \), we use the vector \(\begin{pmatrix} 3 \\ 2 \end{pmatrix}\).
Key Takeaway: For \( y = f(x + a) + b \), the translation vector is \(\begin{pmatrix} -a \\ b \end{pmatrix}\). Notice the sign flip for the \(x\) part!
3. Stretches (Resizing the Graph)
A Stretch pulls the graph away from an axis or squashes it toward it. Every point's distance from the axis is multiplied by a scale factor.
Vertical Stretch: \( y = a f(x) \)
This is outside the bracket, so it affects the \(y\)-coordinates. It is "honest."
- \( y = 3f(x) \): A vertical stretch, scale factor 3. The graph gets 3 times taller.
- \( y = \frac{1}{2}f(x) \): A vertical stretch, scale factor \( \frac{1}{2} \). The graph gets squashed to half its height.
Note: The \(x\)-axis (where \( y=0 \)) stays exactly where it is. Only the points above or below it move.
Horizontal Stretch: \( y = f(ax) \)
This is inside the bracket, so it affects the \(x\)-coordinates. It is "rebellious," so we do the reciprocal (flip the fraction) for the scale factor.
- \( y = f(2x) \): Horizontal stretch, scale factor \( \frac{1}{2} \). The graph is squashed horizontally!
- \( y = f(\frac{1}{3}x) \): Horizontal stretch, scale factor 3. The graph is pulled wider.
Don't worry if this seems tricky! Just remember: if it's inside the \(x\) bracket, "divide" instead of multiply. If you see \( 2x \), you divide all your \(x\)-coordinates by 2.
Key Takeaway: Vertical stretch \( a f(x) \) has scale factor \( a \). Horizontal stretch \( f(ax) \) has scale factor \( \frac{1}{a} \).
4. Step-by-Step: How to Sketch a Transformation
If you are given a original graph and asked to sketch a single transformation, follow these steps:
- Identify the type: Is it a translation (adding/subtracting) or a stretch (multiplying)?
- Identify the direction: Is the change inside the bracket (\(x\)/horizontal) or outside (\(y\)/vertical)?
- Pick key points: Look at the original "turning points" (peaks and valleys) or where the graph crosses the axes (\(x\) and \(y\) intercepts).
- Apply the rule:
- If \( f(x) + a \): Add \(a\) to all \(y\)-coordinates.
- If \( f(x + a) \): Subtract \(a\) from all \(x\)-coordinates.
- If \( a f(x) \): Multiply all \(y\)-coordinates by \(a\).
- If \( f(ax) \): Divide all \(x\)-coordinates by \(a\).
- Sketch: Draw the new shape through your new points.
Did you know? The shape of a satellite dish or a car headlight is a parabola. Engineers use graph transformations to calculate exactly how deep or wide that dish needs to be to reflect signals perfectly!
5. Common Mistakes to Avoid
Even the best mathematicians make these slips! Keep an eye out for them:
- The Sign Flip: Moving \( y = f(x + 3) \) to the right instead of the left. Remember: Inside = Opposite!
- Scale Factor Confusion: Saying \( y = f(2x) \) has a scale factor of 2. It's actually \( \frac{1}{2} \).
- Confusing Axes: Applying a vertical change to the \(x\)-coordinates. Always double-check: Is it outside (up/down) or inside (left/right)?
- Incomplete Descriptions: When asked to "describe" a transformation, you must use the specific math words: "Translation by the vector..." or "Stretch with scale factor... parallel to the... axis."
Summary Checklist
Quick Review Box:
1. \( f(x) + a \) : Vertical Translation (Up/Down)
2. \( f(x + a) \) : Horizontal Translation (Left/Right - watch the sign!)
3. \( a f(x) \) : Vertical Stretch (Scale Factor \(a\))
4. \( f(ax) \) : Horizontal Stretch (Scale Factor \( \frac{1}{a} \))
Top Tip: If you get stuck, try plugging in a simple number. If you aren't sure if \( f(x-2) \) moves left or right, pick a point like \(x=0\) and see what happens to the value!