Skip to content# Transformation of Graphs: A Comprehensive Guide with Worksheet

## Horizontal Shift: f(x + a) and f(x - a)

## Vertical Shift: f(x) + a and f(x) - a

## Vertical Stretching/Shrinking: af(x) and 1/a f(x)

## Horizontal Stretching/Shrinking: f(ax) and f(x/a)

## Reflection: -f(x) and f(-x)

## Conclusion

### Practice Questions: Transformation of Graphs

#### Solutions:

**Transformation of Graphs**

- Graphs are an essential tool in studying functions and their behaviour.
- Transforming the graph of a function involves changing its position or shape without changing the original function.
- These transformations are achieved through various rules that apply to specific functions.

In this article, we will discuss:

**Horizontal Shift: f(x + a) and f(x – a)****Vertical Shift: f(x) + a and f(x) – a****Vertical Stretching/Shrinking: af(x) and 1/a f(x)****Horizontal Stretching/Shrinking: f(ax) and f(x/a)****Reflection: -f(x) and f(-x)**

Here is one more link to practice a few extra questions: Maths Genie Transformation of Graphs Questions

When we shift a graph horizontally, we move it left or right along the x-axis.

The function graph transformation rules for horizontal shifts are as follows:

**f(x + a) shifts the graph of f(x) left by a units****f(x – a) shifts the graph of f(x) right by a units**

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**Solved Example: **

Consider the graph of the function f(x) = x².

If we apply the transformation rule f(x + 2), the new graph will shift two units to the left, resulting in the graph of the function f(x + 2) = (x + 2)².

- A vertical shift moves the graph up or down along the y-axis.

The function graph transformation rules for vertical shifts are as follows:

**f(x) + a vertically shifts the graph of f(x) upward by a units****f(x) – a vertically shifts the graph of f(x) downwards by a units**

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**Solved Example: **

Let’s say we have the graph of the function f(x) = x³.

If we apply the transformation rule f(x) + 4, the new graph will shift four units up, resulting in the graph of the function f(x) + 4 = x³ + 4.

Vertical stretching or shrinking of a graph changes the height of the graph.

The function graph transformation rules for vertical stretching/shrinking are as follows:

**af(x) vertically stretches the graph of f(x) by a factor of a units****1/a f(x) vertically shrinks the graph of f(x) by a factor of a units**

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**Solved Example: Transformation of Graphs**

Let’s consider the graph of the function f(x) = x².

If we apply the transformation rule 2f(x), the new graph will stretch vertically by a factor of 2, resulting in the graph of the function 2f(x) = 2x².

If we apply the transformation rule f(x)/2, the new graph will shrink vertically by a factor of 2, resulting in the graph of the function f(x)/2 = (1/2)x².

Horizontal stretching or shrinking of a graph changes the width of the graph.

The function graph transformation rules for horizontal stretching/shrinking are as follows:

**f(ax) horizontally shrinks the graph of f(x) by a factor of a units****f(x/a) horizontally stretches the graph of f(x) by a factor of a units**

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**Solved Example: Transformation of Graphs**

Let’s say we have the graph of the function f(x) = cos(x).

If we apply the transformation rule f(2x), the new graph will shrink horizontally by a factor of 2, resulting in the graph of the function f(2x) = cos(2x).

If we apply the transformation rule f(x/2), the new graph will stretch horizontally by a factor of 2, resulting in the graph of the function f(x/2) = cos(x/2).

Reflections of a graph involve flipping the graph across the x or y-axis.

The function graph transformation rules for reflections are:

**Reflection across the x-axis: replace f(x) with -f(x) which reflects the graph of f(x) across the x-axis.****Reflection across the y-axis: replace x with -x in f(x) which reflects the graph of f(x) across the y-axis.**

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**Solved Example: Transformation of Graphs**

Consider the function f(x) = x^{2}. Here are some examples of its transformations:

- f(x + 2) shifts the graph left by 2 units.
- f(x – 3) shifts the graph right by 3 units.
- f(x) + 4 shifts the graph upwards by 4 units.
- f(x) – 1 shifts the graph downwards by 1 unit.
- 2f(x) vertically stretches the graph by a factor of 2.
- 1/3f(x) vertically shrinks the graph by a factor of 3.
- f(2x) horizontally shrinks the graph by a factor of 2.
- f(1/2x) horizontally stretches the graph by a factor of 2.
- -f(x) reflects the graph over the x-axis.
- f(-x) reflects the graph over the y-axis.

- Note that you can apply multiple transformations to a single function.
- For example, f(2x – 3) represents a horizontal shrink by a factor of 2 and a horizontal shift right by 3 units.

- In conclusion, understanding the rules of function graph transformation is crucial for graphing and analysing functions.
- With practice and understanding, you will be able to easily recognize and apply these transformations to any given function.

**Question 1:** Consider the function f(x) = x^{2}. Describe the transformation that occurs when f(x) is replaced by f(x + 3).

**Question 2:** Consider the function g(x) = x^{3}. Describe the transformation that occurs when g(x) is replaced by g(x – 2).

**Question 3:** Describe the transformations necessary to transform the graph of f (x) into that of g(x).

**f(x) = x ^{3}**

**g(x) = (x – 3) ^{3 }+ 3**

**Question 4:** Describe the transformations necessary to transform the graph of f (x) into that of g(x).

**f(x) = x ^{3}**

**g(x) = (x + 1) ^{3 }+ 2**

**Question 5:** Describe the transformations necessary to transform the graph of f (x) into that of g(x).

**f(x) = x ^{2}**

**g(x) = (x + 3) ^{3 }– 2**

**Question 6:** Describe the transformations necessary to transform the graph of f (x) into that of g(x).

**f(x) = x ^{3}**

**g(x) = -(x – 2) ^{3}**

**Question 7:** Describe the transformations necessary to transform the graph of f (x) into that of g(x).

**f(x) = x ^{2}**

**g(x) = (x + 1) ^{2 }– 3**

**Question 8:** Describe the transformations necessary to transform the graph of f (x) into that of g(x).

**f(x) = 1/x**

**g(x) = 3/x – 3**

**Question 9:** Describe the transformations necessary to transform the graph of f (x) into that of g(x).

**f(x) = x ^{2}**

**g(x) = -x ^{2 }+ 2**

**Question 10: **Transform the given function f (x) as described and write the resulting function as an equation

**f(x) = x ^{3}**

reflect across the x-axis

translate up 2 units

Question 1: Consider the function f(x) = x^{2}. Describe the transformation that occurs when f(x) is replaced by f(x + 3).

**Solution:**

The transformation f(x + 3) represents a horizontal shift of the graph of f(x) to the left by 3 units.

**Question 2: **Consider the function g(x) = x^{3}. Describe the transformation that occurs when g(x) is replaced by g(x – 2).

**Solution:**

The transformation f(x + 3) represents a horizontal shift of the graph of f(x) to the left by 3 units.

**Question 3:** Describe the transformations necessary to transform the graph of f (x) into that of g(x).

**f(x)=x ^{3}**

**g(x)=(x-3) ^{3}+3**

**Solution:**

translate right 3 units

translate up 3 units

**Question 4:**Describe the transformations necessary to transform the graph of f (x) into that of g(x).

**f(x)=x ^{3}**

**g(x)=(x+1) ^{3}+2**

**Solution:**

translate left 1 unit

translate up 2 units

**Question 5:**Describe the transformations necessary to transform the graph of f (x) into that of g(x).

**f(x)=x ^{2}**

**g(x)=(x+3) ^{3}-2**

**Solution:**

translate left 3 units

translate down 2 units

**Question 6:**Describe the transformations necessary to transform the graph of f (x) into that of g(x).

**f(x)=x ^{3}**

**g(x)=-(x-2) ^{3}**

**Solution:**

reflect across the x-axis

translate right 2 units

**Question 7:** Describe the transformations necessary to transform the graph of f (x) into that of g(x).

**f(x)=x ^{2}**

**g(x)=(x+1) ^{2}-3**

**Solution:**

translate left 1 unit

translate down 3 units

**Question 8:**Describe the transformations necessary to transform the graph of f (x) into that of g(x).

**f(x)=1/x**

**g(x)=3/x – 3**

**Solution:**

expand vertically by a factor of 3

translate down 3 units

**Question 9: **Describe the transformations necessary to transform the graph of f (x) into that of g(x).

**f(x)=x ^{2}**

**g(x)=-x ^{2}+2**

**Solution:**

reflect across the x-axis

translate up 2 units

**Question 10:**Transform the given function f (x) as described and write the resulting function as an equation

**f(x)=x ^{3}**

reflect across the x-axis

translate up 2 units

**Solution:**

g(x)=-x^{3} + 2