## Graph Transformation of Functions

Discussions for the Core part of the syllabus. Algebra, Functions and equations, Circular functions and trigonometry, Vectors, Statistics and probability, Calculus. IB Maths HL Revision Notes

### Graph Transformation of Functions

Horizontal and Vertical Shifts

If $f(x)$ is the original function where $c>0$ then the graph of f(x)+c is shifted up $c$ units,
and the graph of $f(x)-c$ is shifted down $c$ units

A vertical shift means that every point$(x,y)$ on the graph of the original function $f(x)$ is transformed to $(x,y\pm c)$ on the graph of the transformed function $f(x)+c \ or \ f(x)-c$ respectively.

The graph of $f(x+c)$ is shifted left $c$ units

The graph of $f(x-c)$ is shifted right $c$ units

A horizontal shift means that every point$(x,y)$ on the graph of the original function $f(x)$ is transformed to $(x\pm c,y)$ on the graph of the transformed function $f(x-c) \ or \ f(x+c)$ respectively.

Reflections

If $f(x)$ is the original function then

The graph of $-f(x)$ is a reflection in the x-axis.

The graph of $f(-x)$ is a reflection in the y-axis.

Absolute value transformation

$|f(x)|$ : Every part of the graph which is below x-axis is reflected in x-axis.

$f(|x|)$ : For $x \geq 0$ the graph is exactly the same as this of the original function.

For $x <0$ the graph is a reflection of the graph for x≥0 in y-axis.

Stretching and Shrinking

If $f(x)$ is the original function, $c>1$ then

The graph of $cf(x)$ is a vertical stretch by a scale factor of $c$

If $f(x)$ is the original function, $0 then
The graph of $cf(x)$ is a vertical shrink by a scale factor of $c$.

A vertical stretch or shrink means that every point $(x,y)$ on the graph of the original function $f(x)$ is transformed to $(x,cy)$ on the graph of the transformed function $cf(x)$.

If $f(x)$ is the original function, $c>1$ then
The graph of $f(cx)$ is a horizontal shrink by a scale factor of $\frac{1}{c}$.

If $f(x)$ is the original function, $0 then
The graph of $f(cx)$ is a horizontal stretch by a scale factor of $\frac{1}{c}$.

A horizontal stretch or shrink means that every point $(x,y)$ on the graph of the original function $f(x)$ is transformed to $(\frac{x}{c},y)$ on the graph of the transformed function $f(cx)$.

Order of Tranformation
When we perform multiple transformations the order of these transformations may affect the final graph. Therefore we could follow the proposed order (with some exceptions) below to avoid possible wrong final graphs.

1. Horizontal Shifts

2. Stretch / Shrink

3. Reflections

4. Vertical Shifts
elizabeth

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