# Transforming graphs

**Transforming graphs of functions**

In algebra, graphs can be transformed which is carried out by translating, reflecting, stretching or rotating them. In all cases, the equations are amended to make this happen.

For transforming graphs, it is more relevant to express the equations as a function using the function notation.

**Vertical translations**

The following equations represents the below graph:

*y* = *x*², where *y* = *f*(*x*) + 1, *y* = *f*(*x*) + 4, *y* = *f*(*x*) – 3, *y* = *f*(*x*) – 7.

This is therefore represented as *y* = *f*(*x*) + *a*

If *a *is a positive, the curve moved upwards, and if *a* is a negative, then the curve moves downwards*.*

**Horizontal translations**

The following equations represents the below graph:

*y* = *x*² – 3, where *y* = *f*(*x – 1*), *y* = *f*(*x *– 4), *y* = *f*(*x +* 2), *y* = *f*(*x +* 3).

This is therefore represented as *y* = *f*(*x – a*)

If *a* is a positive number, then the curve is translated as *a* units to the left and if *a* is negative then the curve is translated as *a* units to the right.

**Reflections in the ***x*-axis

*x*-axis

The following equations represents the below graph:

*y* = *x²* –2*x* – 2, where *y* = –*f*(*x*), *y* = *f*(*x*).

This graph therefore shows that *y* = –*f*(*x *) is a reflection of *y* = *f*(*x*)

**Reflections in the ***y*-axis

*y*-axis

The following equations represents the below graph:

*y* = *x³* +4*x²* – 3 where *y* = *f*(-*x*), *y* = *f*(*x*).

This graph therefore shows that *y* = *f*(–*x *), is a reflection of *y* = *f*(*x*) in the *y*-axis.

**Stretches in the ***y*-direction

*y*-direction

The following equations represents the below graph:

*y* =* **x²*, where *y* = 2*f*(*x*), *y* = a*f*(*x*),

This graph therefore shows that the distance from the *x*-axis to the curve *y* = *f*(2*x*) is always double the distance from the x-axis to the curve *y* = *f*(*x*).

**Stretches in the ***x*-direction

*x*-direction

The following equations represents the below graph:

*y* = *x*² + 3*x* – 4, where *y* = *f*(2*x*), *y* = *f*(½*x*)

This graph therefore shows that the distance from the *y*-axis to the curve *y* = *f*(2*x*) is always half the distance from the y-axis to the curve *y* = *f*(*x*).