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1.6 Stretches

This document discusses stretches, which change the size of an object or function but not its orientation. It provides examples of vertical stretches about the x-axis, where the y-values are multiplied by a constant factor, horizontal stretches about the y-axis, where the x-values are multiplied by a factor, and stretches that change both x and y-values. Examples show how to determine the equation of an image function when a pre-image function is stretched. Stretches are illustrated through examples of sketching pre-image and image graphs on a grid.

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49 views7 pages

1.6 Stretches

This document discusses stretches, which change the size of an object or function but not its orientation. It provides examples of vertical stretches about the x-axis, where the y-values are multiplied by a constant factor, horizontal stretches about the y-axis, where the x-values are multiplied by a factor, and stretches that change both x and y-values. Examples show how to determine the equation of an image function when a pre-image function is stretched. Stretches are illustrated through examples of sketching pre-image and image graphs on a grid.

Uploaded by

Ashley Elliott
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
Available Formats
Download as PDF, TXT or read online on Scribd
You are on page 1/ 7

MHF4U1  

  Unit  1  Lesson  6  
 

Stretches

Learning Goal: To determine the effects of the following:

•   horizontal stretches about the y-axis,


•   vertical stretches about the x-axis, and
•   stretches about both the x-axis and the y-axis.

Recall: A transformation changes one or more of the location, shape, size, or


orientation of an object in the coordinate plane.

A translation changes the location of an object or function in the coordinate plane but
left all other attributes the same.

A reflection in the x-axis or y-axis changed the orientation and location of the object or
function in the coordinate plane but left all other attributes the same.

Properties of Stretches

The size of an object or shape of a function is changed as a result of a stretch.

The orientation is not changed.

  1  
MHF4U1     Unit  1  Lesson  6  
 
Vertical stretches about the x-axis

Example 1
Consider the function y=f(x). Five specific points on the function are shown. Sketch the
graph g(x)=3f(x).

Example 2

Consider the function y=f(x). Five specific points on the function are shown.
&
Sketch the graph 𝑔(𝑥) = ' 𝑓(𝑥).

Vertical stretches about the x-axis

  2  
MHF4U1     Unit  1  Lesson  6  
 
Example 3

If 𝑓(𝑥) = √𝑥 and 𝑔(𝑥) = 2𝑓(𝑥), then 𝑔(𝑥) = 2√𝑥    is a vertical stretch of f(x) about the x-
axis by a factor of 2.

Horizontal stretches about the y-axis

Example 4

Consider the function y=f(x). Five specific points on the function are shown.
Sketch the graph g(x)=f(2x).

  3  
MHF4U1     Unit  1  Lesson  6  
 
Horizontal stretches about the y-axis
For 𝑦 = 𝑓(𝑥), 𝑖𝑓  𝑔(𝑥) = 𝑓(𝑘𝑥) and |𝑘| > 1, then g(x) is a horizontal compression
&.
of f(x) about the y-axis by a factor of 4

Example 6

If 𝒇(𝒙) = √𝒙    and g(x)=f(2x), then 𝒈(𝒙) = √𝟐𝒙 is a horizontal compression


𝟏
of f(x) about the y-axis by a factor of 𝟐.

Since A(0,0) is on the y-axis, A(0,0) is an invariant point.


The distance of any other image point from the y-axis is half the distance of the
corresponding pre-image point from the y-axis.
C′ is 2 units from the y-axis and C is 4 units from the y-axis.

For y=f(x), if g(x)=f(kx) and 𝟎 < |𝒌| < 𝟏, then g(x) is a horizontal stretch
𝟏.
of f(x) about the y-axis by a factor of 𝒌

  4  
MHF4U1     Unit  1  Lesson  6  
 
Sketching

Example 8

Given the relation 𝑦 = 𝑓(𝑥) shown on the grid,


&
sketch 𝑦 = ' 𝑓(𝑥).

Example 9

Given the relation 𝑦 = 𝑓(𝑥) shown


&
on the grid, sketch 𝑦 = 𝑓(= 𝑥).

Example 10

Given the relation 𝑦 = 𝑓(𝑥) shown on the


grid, sketch 𝑦 = 2𝑓(4𝑥).

  5  
MHF4U1     Unit  1  Lesson  6  
 
Determine the Equation

Example 11

The graph shows two functions, the pre-


image f(x)=|x| and image g(x).
Determine an equation for g(x) in terms
of f(x).

Option 1

1.   (0,0) is an invariant. It could be a vertical stretch.


2.   Draw a few vertical lines from the x-axis to the pre-image and image.

The table shows several pre-image points and the


corresponding image points.

  6  
MHF4U1     Unit  1  Lesson  6  
 

Option 2

1.   The origin (0,0) is invariant It could be a horizontal


stretch about the y-axis.
2.   Draw a few horizontal lines from the y-axis to the pre-
image and image.

Example 12

The function 𝑓(𝑥) = 2? is stretched vertically about the x-axis by a factor of 3 and
compressed horizontally about the y-axis by a
&
factor of '

Determine the equation of the image and sketch


it.

  7  

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