Introduction to Graph Transformations

Imagine you have a photograph on your phone. You can slide it across the screen, flip it horizontally or vertically, or stretch it to make it taller or wider. In Mathematics, transformations do the exact same thing to the graphs of functions!

By understanding a few simple rules, you can take a basic "parent" function (like \(y = x^2\) or \(y = \sin(x)\)) and predict exactly what its more complex version will look like. This is a vital skill for both Paper 1 (where you can't use a calculator) and Paper 2.

1. Translations (The "Slides")

A translation moves every point on a graph a fixed distance in a specific direction. It changes the position of the graph but not its shape or orientation.

Vertical Translation

\(y = f(x) + k\)

  • If \(k > 0\), the graph moves up by \(k\) units.
  • If \(k < 0\), the graph moves down by \(k\) units.
  • Effect on coordinates: \((x, y) \to (x, y + k)\).

Horizontal Translation

\(y = f(x - h)\)

  • If \(h > 0\), the graph moves right by \(h\) units.
  • If \(h < 0\), the graph moves left by \(h\) units.
  • Quick Tip: Horizontal transformations are often "counter-intuitive." If you see \(f(x - 3)\), it actually moves in the positive direction (right). If you see \(f(x + 3)\), it moves in the negative direction (left).
  • Effect on coordinates: \((x, y) \to (x + h, y)\).

Key Takeaway: Vertical changes happen outside the function and affect \(y\). Horizontal changes happen inside the brackets and affect \(x\).

2. Reflections (The "Flips")

Reflections create a mirror image of the graph across an axis.

Reflection in the \(x\)-axis

\(y = -f(x)\)

  • The graph flips upside down.
  • All positive \(y\)-values become negative, and vice versa.
  • Effect on coordinates: \((x, y) \to (x, -y)\).

Reflection in the \(y\)-axis

\(y = f(-x)\)

  • The graph flips sideways.
  • The left side moves to the right, and the right side moves to the left.
  • Effect on coordinates: \((x, y) \to (-x, y)\).

3. Stretches (The "Resizing")

Stretches pull the graph away from an axis or squash it toward an axis. We use a scale factor to describe how much the graph changes.

Vertical Stretch

\(y = p \cdot f(x)\)

  • The graph is stretched vertically by a scale factor of \(p\).
  • Points on the \(x\)-axis (where \(y = 0\)) stay exactly where they are. These are called invariant points.
  • Effect on coordinates: \((x, y) \to (x, py)\).

Horizontal Stretch

\(y = f(qx)\)

  • The graph is stretched horizontally by a scale factor of \(\frac{1}{q}\).
  • Watch out! This is the most common mistake. If you see \(y = f(2x)\), the graph is actually squashed (compressed) by a factor of \(\frac{1}{2}\). If you see \(y = f(\frac{1}{3}x)\), the graph is stretched wider by a factor of \(3\).
  • Points on the \(y\)-axis stay where they are.
  • Effect on coordinates: \((x, y) \to (\frac{x}{q}, y)\).

Quick Review Box:
Outside the function \(f(x)\) = Vertical change (logical).
Inside the function \(f(x)\) = Horizontal change (reversed/reciprocal).

4. Composite Transformations

Often, you will have to perform more than one transformation at once. For example, \(y = 2f(x - 3) + 1\).

The Order Matters! Generally, you should follow the order of operations (BIDMAS/PEMDAS):

  1. Horizontal translations.
  2. Stretches and Reflections.
  3. Vertical translations.

Note for SL Students: You are not required to transform functions in the form \(f(ax + b)\). You will only deal with cases where the horizontal translation and stretch are separate steps.

5. AHL (Higher Level Only) Transformations

If you are an HL student, you need to master several additional types of transformations.

Modulus Transformations

1. \(y = |f(x)|\)
Any part of the graph that is below the \(x\)-axis (where \(y\) is negative) is reflected upward so it becomes positive. The parts already above the \(x\)-axis stay the same.

2. \(y = f(|x|)\)
First, ignore the part of the graph where \(x < 0\) (the left side). Then, take the part where \(x \geq 0\) (the right side) and mirror it onto the left side. The resulting graph will always be symmetrical about the \(y\)-axis (it becomes an even function).

Reciprocal Transformation: \(y = \frac{1}{f(x)}\)

This is a challenging transformation. Keep these rules in mind:

  • Where \(f(x) = 0\), the new graph has a vertical asymptote.
  • Where \(f(x)\) has a vertical asymptote, the new graph has a zero (intercept).
  • Local maximums on \(f(x)\) become local minimums on \(\frac{1}{f(x)}\), and vice versa.
  • If \(f(x)\) is very large, \(\frac{1}{f(x)}\) is very small (close to the \(x\)-axis).

The Composite \(f(ax + b)\)

When multiple horizontal transformations occur, you must be careful. It is best to factorize the inside: \(f(a(x + \frac{b}{a}))\).

  1. Horizontal stretch by factor \(\frac{1}{a}\).
  2. Horizontal translation by \(\frac{b}{a}\) units.

Power Transformation: \(y = [f(x)]^n\)

This involves raising the \(y\)-value of every point to the power of \(n\). Common examples include squaring the function (\(y = [f(x)]^2\)). Notice that if \(n\) is even, all \(y\)-values will become non-negative.

Common Mistakes to Avoid

  • Mixing up \(h\) and \(k\): Remember that the number inside the parenthesis with \(x\) moves it left/right, and the number outside moves it up/down.
  • Scale Factor Confusion: In \(f(2x)\), the scale factor is not \(2\); it is \(\frac{1}{2}\). Always use the reciprocal for horizontal stretches.
  • Order of Operations: Doing a vertical translation before a vertical stretch will result in a different graph. Usually, you stretch first, then shift!

Did you know? Transformations are used in computer graphics and animation. When a character moves across a screen or shrinks into the distance, the computer is simply applying translation and stretch matrices to the functions defining that character's shape!