Transforming Graphs of Functions

The Two Golden Rules of Graph Transformations

At the heart of all graph transformations are two simple, powerful rules that determine how a change to the function's equation affects the graph. Understanding these will allow you to decode any transformation question an examiner throws at you.

The key is to notice where the change is happening in the function notation: is it inside the bracket with the $x$, or outside the bracket?

1. Inside the Bracket (Affects X): Any change made inside the function's bracket, directly affecting the $x$ variable (e.g., $f(x+a)$, $f(ax)$, $f(-x)$), transforms the graph horizontally. It only changes the x-coordinates. Crucially, the effect is the opposite of what you might expect.
2. Outside the Bracket (Affects Y): Any change made outside the function's bracket (e.g., $f(x)+a$, $af(x)$, $-f(x)$) transforms the graph vertically. It only changes the y-coordinates. This effect is direct and does exactly what you'd expect.

Let's watch a short video clip that introduces these fundamental rules.

Now, let's break down the specific types of transformations using these two rules.

1. Translations (Shifting the Graph)

A translation simply slides the entire graph without changing its shape or orientation.

• Vertical Translation: $y=f(x)+a$

  • The + a is outside the bracket.
  • It affects the y-coordinates as you'd expect.
  • If $a$ is positive, the graph shifts up by $a$ units.
  • If $a$ is negative, the graph shifts down by $a$ units.
  • Effect on a point (x, y): $(x,y)\to (x,y+a)$

• Horizontal Translation: $y=f(x+a)$

  • The + a is inside the bracket.
  • It affects the x-coordinates in the opposite way.
  • $y=f(x+a)$ shifts the graph left by $a$ units (opposite of plus).
  • $y=f(x-a)$ shifts the graph right by $a$ units (opposite of minus).
  • Effect on a point (x, y): $(x,y)\to (x-a,y)$

2. Stretches (Scaling the Graph)

A stretch pulls or squashes the graph, changing its shape.

• Vertical Stretch: $y=af(x)$

  • The multiplier a is outside the bracket.
  • It affects the y-coordinates as you'd expect.
  • The graph is stretched vertically by a scale factor of $a$. All y-coordinates are multiplied by $a$.
  • Effect on a point (x, y): $(x,y)\to (x,ay)$

• Horizontal Stretch: $y=f(ax)$

  • The multiplier a is inside the bracket.
  • It affects the x-coordinates in the opposite way.
  • The graph is stretched horizontally by a scale factor of $\frac{1}{a}$. All x-coordinates are multiplied by $\frac{1}{a}$ (the reciprocal of $a$).
  • Effect on a point (x, y): $(x,y)\to (\frac{x}{a},y)$

3. Reflections (Flipping the Graph)

A reflection flips the graph across an axis. This is a special case of stretching where the scale factor is -1.

• Reflection in the x-axis: $y=-f(x)$

  • The negative sign is outside the bracket.
  • It affects the y-coordinates as you'd expect (multiplies them by -1).
  • Effect on a point (x, y): $(x,y)\to (x,-y)$

• Reflection in the y-axis: $y=f(-x)$

  • The negative sign is inside the bracket.
  • It affects the x-coordinates. For reflections, the "opposite" rule can be confusing; it's easier to just remember it flips the sign of the x-coordinate.
  • Effect on a point (x, y): $(x,y)\to (-x,y)$

This summary table and poster are excellent references to keep handy.

Applying the Rules: From Points to Sketches

In your exam, you'll be asked to do two main things: find the new coordinates of a specific point, or sketch the entire transformed graph.

Let's watch how to apply these rules step-by-step. The video demonstrates how to transform a single maximum/minimum point and then uses that skill to transform an entire curve.

Sketching Transformed Graphs

The method for sketching a transformed graph is a logical extension of transforming a single point.

1. Identify key anchor points on the original graph. These are usually the y-intercept, x-intercepts, and any maximum or minimum points.
2. Apply the transformation rules to each of these anchor points to find their new positions.
3. Draw a smooth curve through the new anchor points, making sure the shape of the new graph is consistent with the transformation (e.g., stretched, shifted, or reflected).

Exam Practice

Now it's time to apply these methods, as you would in an exam. This is the core of the "Error-Loop Method" — putting theory into practice.

Conclusion

Today you've learned a systematic, rule-based approach to graph transformations. By remembering the two golden rules—inside the bracket affects X (opposite), outside affects Y (expected)—you can confidently tackle questions on translations, stretches, and reflections.

Key Takeaways:

• Translations (Shifts): $f(x)+a$ is vertical, $f(x+a)$ is horizontal (in the opposite direction).
• Stretches (Scales): $af(x)$ is a vertical stretch by factor $a$, $f(ax)$ is a horizontal stretch by factor $\frac{1}{a}$.
• Reflections (Flips): $-f(x)$ reflects in the x-axis, $f(-x)$ reflects in the y-axis.
• Method for Exam Questions:
  1. Identify the transformation(s) from the equation.
  2. Apply the rules one by one to find the new coordinates of key points.
  3. If sketching, connect the new points with a smooth curve that respects the transformation.

Preview of the Next Lesson:
In our final lesson of this module, we will apply your graph interpretation skills to a real-world context: kinematics. You will learn how to calculate speed, acceleration, and distance from distance-time and speed-time graphs. This bridges the gap from abstract functions to the practical application of graphs in physics and other sciences.