Translations and Enlargements of Graphs Using Functions

Graphs of functions can be transformed in different ways, such as translations (shifting the graph) and enlargements (stretching or shrinking the graph). These transformations affect the function’s equation in predictable ways.

 

Translations (Shifting the Graph)

• A translation moves the graph without changing its shape. This can be done horizontally or vertically.
• These translations can be represented in vector form or in function form. 
• In vector form for example $\begin{pmatrix} a \\ b \end{pmatrix}$ means translate the $x-axis$ by $a$ and the $y-axis$ by $b$

 

 

Vertical Translation: $f(x) + k$ or $\begin{pmatrix}  + k \\ 0 \end{pmatrix}$

• The whole graph moves up by $k$ if $k > 0$ or $\begin{pmatrix} +k  \\ 0 \end{pmatrix}$
• The whole graph moves down by $k$ if $k < 0$ or $\begin{pmatrix} - k  \\ 0 \end{pmatrix}$
• The shape remains the same, only the position changes

Example

Sketch the graphs of $g(x), \text{Where } k = 3, \ i.e f(x) + 3$ and $h(x) , \text{Where } k = - 3, \ i.e f(x) - 3$ given $f(x) = x^2$ 

 

 

Horizontal Translation: $f(x + h)$ or $\begin{pmatrix} + h  \\ 0 \end{pmatrix}$

• The graph moves to the right by $h$ if $h < 0$ or $\begin{pmatrix} +h  \\ 0 \end{pmatrix}$
• The graph moves to the left by $h$ if $h > 0$ or $\begin{pmatrix} - h  \\ 0 \end{pmatrix}$
• As you can see horizonal transformations do the opposite of what you would think would happen based on their value

 

Example

Sketch the graphs of $g(x) = (x - 4)^2$ and $h(x) = (x + 3)^2$ given $f(x) = x^2$ 

Solution

• Notice how $g(x)$ and $h(x)$ relate to $f(x)$. $g(x)$ is essentially $f(x)$ if instead of the input being $x$, the input is $x - 4$.
• This shows that \(g(x) = \(f(x - 4)\), showing that the translation is to the right by $4$
• The same can be done for $h(x)$. This would give $h(x) = f(x + 3)$, meaning the graph would move to the left by $3$

 

 

 

 

Enlargements (Stretching or Shrinking the Graph)

Enlargements i.e stretching or shrinking the graph involve changing the relative size of the graph in either the x or y plane

 

Vertical Stretch or Compression: $af(x)$

• A vertical stretch can change the graph in different ways depending on the value of a
• If $a > 1$ the graph gets taller and is stretched.
• If $0< a < 1$ or $-1< a < 0$ the graph gets shortened and is compressed. This is as the scale factor is a fraction
• If $a$ is a negative then the graph is also flipped/reflected in the $x-\text{axis}$
• We can perform these transformations by multiplying all the y-coordinates by $a$

 

Example

Sketch the graphs of $g(x) = 2x^2$ and $h(x) = \frac{1}{2}x^2$ given $f(x) = x^2$ 

 

 

 

Horizontal Stretch or Compression: $f(bx)$

• A vertical stretch can change the graph in different ways depending on the value of b.
• Horizontal enlargements behave opposite to how you would think they would.
• If $b > 1$ the graph gets compressed
• If $0< b < 1$ or $-1< b < 0$ the graph gets stretched.
• If $b$ is a negative then the graph is also flipped/reflected in the $y-\text{axis}$
• We can perform these transformations by multiplying all the x-coordinates by $\frac{1}{b}$

 

Example

Sketch the graphs of $g(x) =  f(\frac{x}{2}) =  (\frac{x}{2})^2$ and $h(x) =  f(3x) = (3x)^2$ given $f(x) = x^2$