Transforming Functions: Shifts

1. Vertical Translations (Up and Down)

A vertical translation, or shift, moves every point on a graph up or down by the same amount. The shape of the graph remains identical.

This is the more intuitive of the two translations we'll study today. Let's say we have a function f(x). To shift its graph vertically, we simply add a constant to the entire function's output.

The rule is: y = f(x) + c

• If c is positive, the graph shifts up by c units.
• If c is negative, the graph shifts down by c units.

Think of it this way: for any given x value, you calculate f(x) and then simply add c to the final y-coordinate. This moves the point vertically.

To see this in action, let's watch a short video that uses dynamic graphing software to make the concept very clear.

For a text-based summary, the following resource is useful. It also introduces the concept of describing translations using vectors, which is a neat way to formalise the movement. A shift of c units up is represented by the vector (0, c).

2. Horizontal Translations (Left and Right)

A horizontal translation shifts a graph left or right. The rule for this can seem a bit counter-intuitive at first, so it's important to pay close attention.

The rule is: y = f(x + c)

• If c is positive (e.g., f(x + 2)), the graph shifts to the left by c units.
• If c is negative (e.g., f(x - 2)), the graph shifts to the right by c units.

Why is this counter-intuitive?
We naturally associate + with moving right and - with moving left. The key is to remember that the change is happening inside the function, to the x input itself.

Consider f(x - 2). To get the same y-value that f(x) produced at x=0 (i.e., f(0)), we now need to plug in x=2 into our new function: f(2 - 2) = f(0). So the point that was at x=0 has moved to x=2—a shift to the right.

Let's continue with the same video, as it does a great job of explaining and visualising this "backwards" behaviour.

The "Common misconceptions" section in the Graph Transformations article also provides a good written explanation for why this happens. I recommend reviewing it to solidify your understanding.

This image provides a fantastic summary of the rules for both vertical and horizontal translations. It's a great reference to keep in mind.

3. Combining Translations and Applying to Points

Now, we can put both types of translation together. A function can be transformed both horizontally and vertically in the same step. The general form for this is:

$y=f(x-h)+k$

• The + k outside the function causes a vertical shift (up for +k, down for -k).
• The - h inside the function causes a horizontal shift (right for +h, left for -h).

This means the entire graph of f(x) is shifted so that the point that was at (0,0) on the original graph moves to (h, k).

A common way to test this knowledge is to ask what happens to a single point on the graph after a transformation. This is often simpler than sketching the whole new graph.

This next video is perfect for seeing how to handle combined transformations and how to apply them to a single point.

Let's work through an example using these rules.
Suppose the point P(3, 7) is on the graph of y = f(x). We want to find where P moves to on the graph of y = f(x - 2) + 4.

1. Analyse the transformation:
   • x - 2 inside the brackets means a horizontal shift 2 units to the right.
   • + 4 outside the brackets means a vertical shift 4 units up.
2. Apply to the coordinates:
   • The x-coordinate moves 2 to the right: 3 + 2 = 5.
   • The y-coordinate moves 4 up: 7 + 4 = 11.
3. The new point is (5, 11).

Conclusion

In this lesson, we've developed a set of rules for translating the graph of any function. This is a powerful tool that allows us to understand a whole family of graphs based on a single "parent" function.

Key Takeaways:

• Vertical Translations: y = f(x) + c shifts the graph. It moves up if c is positive and down if c is negative. This is intuitive.
• Horizontal Translations: y = f(x + c) shifts the graph. It moves left if c is positive and right if c is negative. This is counter-intuitive and a key point to remember.
• Combined Translations: A transformation y = f(x - h) + k shifts the graph h units horizontally and k units vertically.
• These rules can be applied to an entire graph or to a single point on the graph.

Preview of the Next Lesson:
Translations are just one way to transform a function. What if we want to stretch, compress, or flip a graph? In our next lesson, we will cover these other transformations: vertical and horizontal stretches/compressions and reflections. These will complete our toolkit for manipulating function graphs.