Compressions and Stretches

Key Concepts

• Reflections of functions across the “x-axis”.
• Vertical stretches and compressions of graphs of functions.
• Horizontal stretches and compressions of graphs of functions.

Quadratic functions & its characteristics 

The quadratic parent function is 𝒇𝒙=𝒙𝟐

It is the simplest function in quadratic function family. The graph of the function is a curve called a parabola. 

The vertex is the lowest/highest point on the graph of a quadratic function.  

The vertex form of a quadratic function is  𝒇𝒙=𝒂(𝒙−𝒉)𝟐+𝒌 .  

The graph of f is the graph of 𝒈𝒙=𝒂𝒙𝟐 translated horizontally 𝒉 units and vertically 𝒌 units.  

The vertex is located at 𝒉, 𝒌.  

The axis of symmetry is 𝒙=𝒉.

The vertex form of a quadratic function is 𝒇𝒙=𝒂(𝒙−𝒉)𝟐+𝒌. 

The graph of f is the graph of

g(x)=ax2𝒈𝒙=𝒂𝒙𝟐 translated horizontally 𝒉 units and vertically 𝒌 units.  

The vertex is located at 𝒉, 𝒌.  

The axis of symmetry is𝒙=𝒉. 

Translations of functions

Vertical translation  

For a given

g(x)=f(x)+k, the graph of the function g is the function f translates k units vertically.  

k>0: shifts |k|units up  

k<0: shifts |k| units down  

Horizontal translation

For a given

g(x)=f(x−h), the graph of the function g is the function f translates h units horizontally.  

h>0: shifts |h|units right  

h<0: shifts |h| units left 

Combined translation

For a given

g(x)=f(x−h)+k, the graph of the function g is the graph of function f translated h units horizontally, then translated k units vertically. 

Example:  

How does the function

f(x)=x2 transform to the function g(x)=(x−3)2+2.  Also find the vertex of the function g(x). 

Solution:  

Graph of the function

f(x)=x2 and g(x)=(x−3)2+2. 

The function g(x) is of the form (x−h)2+k(x−h)2+k has a translation of h units horizontally and k units vertically.  

The function f(x)=x2 transforms to the function g(x)=(x−3)2+2:  

3 units right horizontally and 2 units up vertically.  

The vertex of the function g(x) is (3, 2). 

Reflections across the x-axis 

Example 1:  

Consider the function

f(x)=√fx=x

and

g(x)=√gx=−x

The graph of g(x)=√x is a reflection of f(x)=√x across the x−axis.  

Example 2:  

Consider the function

f(x) = x²

and

g(x) =−x²

The graph of g(x)=−x² is a reflection of f(x)=x² across the x−axis.  

In general, g(x)=−1f(x) , the graph of g is a reflection across the x−axis of the graph of f.  

So, for any function when the output is multiplied by −1 it reflects across the x−axis.   

Vertical compressions and stretches of graph

Vertical stretches of graphs

Example:  

Consider g(x)=kf(x) for |k|>1k>1 when function f(x)=x². 

Solution:   

Here is the graph of g is a vertical stretch away from the x−axis  of the graph of f. 

So, the graph of g(x)=2x² is vertical stretch of f(x)=x² away from the x−axis. 

Example:  

Consider g(x)=kf(x) for |k|>1k>1 when function fx=x³

Solution:  

Here is the graph of g is a vertical stretch away from the x−axis  of the graph of f. 

So, the graph of g(x)=2x³ is vertical stretch of f(x)=x³ away from the x−axis. 

Vertical compressions of graphs

Example:  

Consider g(x)=kf(x) for 0<|k|<1 when function f(x)=|x+1|. 

Solution:   

Here is the graph of g is a vertical compression towards the x−axis of the graph of f. 

So, the graph of g(x)=1/2 |x+1| is vertical compression of f(x)=|x+1| towards the x−axis. 

Example:  

Consider g(x)=kf(x) for 0<|k|<1 when function f(x)=x².  

Solution:    

Here is the graph of g is a vertical compression towards the x−axis of the graph of f. 

So, the graph of g(x)=1/2 x² is vertical compression of f(x)=x² towards the x−axis.  

Vertical compressions & stretches of graphs

Given a function,

f(x) , a new function g(x)=kf(x), where a is a constant, is vertical stretch or a vertical compression of the function f(x).  

• If |k|>1k>1, then the graph will be stretched.  
• If 0<|k|<10<k<1, then the graph will be compressed. 

Horizontal compressions and stretches of graphs

Horizontal stretches of graphs

Example:  

Consider g(x)=f(kx) for 0<k<1 when function f(x)=x²

Solution:  

Here is the graph of g is a horizontal compression toward the y-axis of the graph of f. 

So, the graph of gx= (1/2 x)² is horizontal compression of fx=x² toward the y-axis. 

Example:  

Consider g(x) = f(kx) for [0<k<1] when function f(x) = √x 

Solution:  

Here is the graph of g is a horizontal compression toward the y-axis of the graph of f. 

So, the graph of g(x)= 1/4 x is horizontal compression of f(x)=√x toward the y-axis. 

Horizontal compressions of graphs

Example:  

Consider g(x)=f(kx) for k>1 when function f(x)=(x−1)²

Solution:  

Here is the graph of g is a horizontal compression toward the y-axis of the graph of f. 

So, the graph of g (x)= (3x-1)² is horizontal compression of f(x)= (x-1)² toward the y-axis. 

Solution:  

Here is the graph of g is a horizontal compression toward the y-axis of the graph of f. 

So, the graph of g(x) = |2x+1| is horizontal compression of f(x) = |x+1| toward the [y-axis]. 

Horizontal compressions & stretches of graphs

Given a function [f(x)], a new function gx = f(kx), where k is a constant, is a horizontal stretch or a horizontal compression of the function f(x).  

• If [|k|<1], then the graph will be compressed. 
• If [0<|k|<1], then the graph will be stretched. 

Exercise

• Write a function with a graph that is the reflection of the graph of fx=(x-1)² across the x-axis
• For each pair, identify the graph of g is a vertical or horizontal compression or stretch of the graph of f.
  • f(x)=|3x-1| , g(x) = 1/4 |3x-1|
  • f(x) = √x , g(x) = √1/5x
  • f(x)=√x-4 , g(x)=√3x-4
  • f(x) = x²+4 , g(x)=2x²+8

• Write a function with a graph that is a vertical stretch of the graph of fx=|x|, away from the x-axis.
• Write a function with a graph that is a horizontal compression of the graph of fx=3x, toward the y-axis.

Concept Summary  

What we have learned

• Reflections across the x-axis.
• Vertical stretches of graphs.
• Vertical compressions of graphs.
• Horizontal stretches of graphs.
• Horizontal compressions of graphs.