International Mathematics (0607) | Transforming graphs of functions

🌟 Transforming Graphs of Functions: Study Notes (IGCSE 0607 Extended) 🌟

Welcome to the exciting world of graph transformations! Don't worry if this sounds complex—it’s actually a fantastic shortcut. Instead of plotting a new table of values every time a function changes slightly, we learn simple rules to *move* the original graph.

Understanding these transformations is crucial because it allows you to quickly sketch and interpret complicated functions based on familiar shapes (like parabolas or straight lines). This skill is particularly useful when working with your Graphic Display Calculator (GDC) to verify your mental sketches.

1. Introduction to Graph Transformations

What is a Transformation?

In mathematics, a graph transformation is simply moving or resizing the graph of a function without changing its basic underlying shape. If you start with a function \(y = f(x)\), any changes you make to the equation will transform the graph.

We call the original function, \(y = f(x)\), the Parent Function. All other related functions are just transformations of this parent.

Did you know?

The term we use for moving a shape without rotating or reflecting it is a Translation. For this chapter, we focus specifically on translating graphs horizontally (left/right) or vertically (up/down).

2. Vertical Translations: Moving Up and Down

A vertical translation occurs when you add or subtract a constant value outside the main function operation.

The Rule: \(y = f(x) + k\)

When you take the original function \(f(x)\) and add an integer constant \(k\) to the whole expression, the graph shifts vertically.

• If \(k\) is positive (e.g., \(+3\)): The graph shifts UP by \(k\) units.
• If \(k\) is negative (e.g., \(-5\)): The graph shifts DOWN by \(|k|\) units.

Think of it this way: For any given \(x\), the new \(y\)-value is simply the old \(y\)-value plus \(k\). This is why the movement is so direct and intuitive.

Step-by-Step Example

Let the parent function be \(f(x) = x^3\). Describe the transformation for \(y = x^3 - 4\).

1. Identify the structure: It is in the form \(y = f(x) + k\), where \(f(x) = x^3\).
2. Identify \(k\): \(k = -4\).
3. Describe the translation: Since \(k\) is negative and outside the function, the graph translates 4 units down.
4. Coordinate change: If a point \((x, y)\) was on \(f(x)\), the corresponding point on the transformed graph is \((x, y - 4)\).

Memory Aid for Vertical Shifts (The "Direct" Rule)

Vertical (Y) shifts are Yes-men. They do exactly what they are told. Plus means up, minus means down.

Key Takeaway for Vertical Shifts: Changes outside the function affect the output (\(y\)-values) and are direct.

3. Horizontal Translations: Moving Left and Right

A horizontal translation occurs when you add or subtract a constant value inside the function, directly affecting the variable \(x\).

The Rule: \(y = f(x + k)\)

When you substitute \((x + k)\) in place of \(x\) in the function \(f(x)\), the graph shifts horizontally. This is the part that can be tricky!

• If \(k\) is positive (e.g., \(f(x+3)\)): The graph shifts LEFT by \(k\) units.
• If \(k\) is negative (e.g., \(f(x-5)\)): The graph shifts RIGHT by \(|k|\) units.

Why is Horizontal Movement Counter-Intuitive?

Imagine you have the function \(f(x)\). To get the original output value, say \(f(0)\), on the new graph \(f(x+2)\), you need to input a different value of \(x\).
To make the bracket equal to zero, we must set \(x+2=0\), so \(x=-2\). This means the point that was originally at \(x=0\) has now moved to \(x=-2\). The graph shifted left!

Step-by-Step Example

Let the parent function be \(f(x) = \frac{1}{x}\). Describe the transformation for \(y = \frac{1}{x-1}\).

1. Identify the structure: It is in the form \(y = f(x + k)\), where the change is inside the function (the denominator).
2. Identify the required shift: We see \((x-1)\). Since this is minus one, the rule is the opposite.
3. Describe the translation: The graph translates 1 unit right.
4. Coordinate change: If a point \((x, y)\) was on \(f(x)\), the corresponding point on the transformed graph is \((x + 1, y)\).

Memory Aid for Horizontal Shifts (The "Inverse" Rule)

Horizontal (X) shifts are X-rated (or eXtra confusing). They do the opposite of what the sign suggests. Plus means left, minus means right.

Key Takeaway for Horizontal Shifts: Changes inside the function affect the input (\(x\)-values) and are inverse (opposite).

4. Identifying and Describing Transformations (Summary)

The core skill required is to look at a new function and describe exactly how it relates to the parent function \(y = f(x)\).

Example Scenarios

Suppose the parent function is \(f(x)\). We are given a new function \(g(x)\) and need to describe the transformation:

1. \(g(x) = f(x) + 7\)
   Description: Translation 7 units up. (Vertical, Direct)
   Vector: \(\begin{pmatrix} 0 \\ 7 \end{pmatrix}\)
2. \(g(x) = f(x - 2)\)
   Description: Translation 2 units right. (Horizontal, Inverse)
   Vector: \(\begin{pmatrix} 2 \\ 0 \end{pmatrix}\)
3. \(g(x) = f(x + 1) - 3\) (Note: Although E3.6 only requires simple single transformations, you must be ready to identify both components if they are combined)
   Description: Translation 1 unit left, and 3 units down.
   Vector: \(\begin{pmatrix} -1 \\ -3 \end{pmatrix}\)

📍 Common Mistakes to Avoid

The number one mistake is mixing up the vertical and horizontal rules. Always remember:

• Outside Change (\(f(x) + k\)): Changes \(y\). Easy and Direct.
• Inside Change (\(f(x + k)\)): Changes \(x\). Confusing and Opposite.