Graph Transformations
Graph transformations allow us to quickly sketch how a graph changes when its equation is adjusted. This powerful skill helps us predict how changing a situation, like launching a rocket from a higher platform, affects its path, without having to plot everything from scratch.
The Four Key Transformations
Vertical Translation.
$y = f(x) + a$
The graph slides vertically.
- If $a$ is positive, it moves up.
- If $a$ is negative, it moves down.
Horizontal Translation.
$y = f(x+a)$
The graph slides horizontally. Remember the “inside is opposite” rule!
- If it’s $(x+a)$, it moves left.
- If it’s $(x-a)$, it moves right.
Reflection in the x-axis.
$y = -f(x)$
The graph flips vertically over the x-axis. All y-coordinates change sign.
Reflection in the y-axis.
$y = f(-x)$
The graph flips horizontally over the y-axis. All x-coordinates change sign.
Worked Examples
Example 1: Vertical Translation.
The graph of $y=x^2$ is shown. Sketch $y=x^2+3$. This is in the form $y=f(x)+a$. The $+3$ is outside the function, so it’s a vertical shift up by 3 units. The turning point at $(0,0)$ moves to $(0,3)$.Example 2: Horizontal Translation
Given $y=x^2$, sketch $y=(x-2)^2$. This is in the form $y=f(x-a)$. The $-2$ is inside the function, so it’s a horizontal shift. Remember “inside is opposite”, so $-2$ means a shift right by 2 units. The turning point at $(0,0)$ moves to $(2,0)$.Tutor Insights
🤔 Common Misunderstandings
- The “Inside is Opposite” Rule: This is the biggest hurdle! Students often assume $f(x+3)$ moves right. Practice is key to remembering it moves left.
- Confusing Reflections: Mixing up $y=-f(x)$ (reflection in x-axis) and $y=f(-x)$ (reflection in y-axis).
📝 Common Exam Mistakes
- Incorrect shift direction for horizontal translations.
- Not labelling key points: You must show the new coordinates of turning points or intercepts after the transformation.
- Drawing a rough sketch that doesn’t accurately represent the shape and key features.
Practice Questions
- Describe fully the transformation that maps the graph of $y=x^2$ onto the graph of $y=(x+3)^2$.
- The point $(4, 1)$ lies on the graph $y=f(x)$. State the coordinates of the corresponding point on the graph $y=-f(x)$.
- The graph of $y=g(x)$ has a turning point at $(-2, 5)$. What are the coordinates of the turning point on the graph of $y=g(x)-4$?
Show Answers
- A translation of 3 units to the left.
- For $y=-f(x)$, the y-coordinate changes sign. The new point is (4, -1).
- This is a vertical translation down by 4 units. The new turning point is (-2, 1).
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