How can you change the location and direction of a linear function?
Translations, reflections, stretches, and shrinks.
Practice makes perfect
The four main types of transformations are: translation, reflection, stretching, and shrinking. Let's look at each of these one by one.
Translations
The translation of a function can either be vertical or horizontal. Let's begin by looking at a fairly basic function, f(x)=2x.
If we want to vertically translate this function up by 2, then we have to take f(x) and add 2.
y=f(x)+2 ⇒ y=2x+2
If we want to horizontally translate this function by 2 to the right, we have to write y=f(x-2).
y=2(x-2) ⇒ y=2x-4
Reflections
The reflection of a function can be across either axis. Let's begin by looking at the function, f(x)=x+1.
A reflection in the x-axis requires us to reverse the sign of each y-value. Therefore, we will write y=- f(x).
y=-(x+1) ⇒ y=- x-1
A reflection in the y-axis requires us to reverse the sign of each x-value. Therefore, we will write y=f(- x).
y=- x+1
Stretches
The stretching of a function can either be vertical or horizontal. Let's begin by looking at the function f(x)=x+2.
If we want to vertically stretch this function by a factor of 3, we will multiply each y-value by 3.
y=3(x+2) ⇒ y=3x+6
If we want to horizontally stretch this function by a factor of 4, we will multiply each x-value by 14.
y=1/4x+2
Shrinks
Similar to stretching, shrinking can either be vertical or horizontal. Let's begin by looking at the same function that we used for the stretching, f(x)=x+2.
If we want to vertically shrink this function by a factor of 13, we will multiply each y-value by 13.
y=1/3(x+2) ⇒ y=1/3x+2/3
If we want to horizontally shrink this function by a factor of 14, we will multiply each x-value by 4.
y=4x+2
