Transforming graphs

​​In a nutshell

You can apply transformations to an unknown function by considering how specific coordinates would be affected by the transformations applied. Even if the function is unknown, function notation can still be used to describe the transformation.

Transforming unknown functions

When a function is unknown and $$f(x)$$​ is not given, it is still possible to transform the function if specific coordinates are provided. This can also help when drawing transformations of unknown functions. 

Example

Points $$A(2,7)$$, $$B(7,12)$$ and $$C(5,16)$$​ lie on this function. 

Write the new values of $$A,\ B$$ and $$C$$ in the following functions:

a: $$y=f(x+1)$$

b: $$y=-f(x)$$

c: $$y=f(x)-1$$

​

a: The tranformation of $$y=f(x+1)$$ translates $$y=f(x)$$​ by the vector $$\begin{pmatrix}-1\\ 0\end{pmatrix}$$​, therefore:

​

$$A:\underline{(1,7)},\ B:\underline{(6,12)},\ C:\underline{(4,16)}$$​​

b: $$y=-f(x)$$ means the function is reflected in the $$x$$-axis, therefore:

​

$$A:\underline{(2,-7)},\ A:\underline{(7,-12)},\ B:\underline{(5,-16)}$$

c: The transformation of $$y=f(x) - 1$$ translates the function $$y=f(x)$$ by the vector $$\begin{pmatrix}0\\ -1\end{pmatrix}$$​, therefore:

​

$$A:\underline{(2,6)},\ B:\underline{(7,11)},\ C:\underline{(5,15)}$$​​