#### Need Help?

Get in touch with us

# 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.

### 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.

#### Composite Figures – Area and Volume

A composite figure is made up of simple geometric shapes. It is a 2-dimensional figure of basic two-dimensional shapes such as squares, triangles, rectangles, circles, etc. There are various shapes whose areas are different from one another. Everything has an area they occupy, from the laptop to your book. To understand the dynamics of composite […] #### Special Right Triangles: Types, Formulas, with Solved Examples.

Learn all about special right triangles- their types, formulas, and examples explained in detail for a better understanding. What are the shortcut ratios for the side lengths of special right triangles 30 60 90 and 45 45 90? How are these ratios related to the Pythagorean theorem?  Right Angle Triangles A triangle with a ninety-degree […] #### Ways to Simplify Algebraic Expressions

Simplify algebraic expressions in Mathematics is a collection of various numeric expressions that multiple philosophers and historians have brought down. Talking of algebra, this branch of mathematics deals with the oldest concepts of mathematical sciences, geometry, and number theory. It is one of the earliest branches in the history of mathematics. The study of mathematical […]   