Wize Grade 11 Mathematics Textbook > Transformations on Functions

Translations

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Translations
Translations
When graphing a function we can keep the shape the same, and change its location.  This is type of transformation is called a translation or a shift of the original function.

Example 1


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Translations from the equation
We can apply a translation to a function by adding or subtracting a positive constant to the original function.
y=f(x)+cy=f(x−d)\begin{aligned}
 y &= f(x) + \colorFour{c} \\
 y &= f(x - \colorThree{d})
\end{aligned}yy​=f(x)+c=f(x−d)​
Vertical Translation
Up if c\colorFour{c}c is positive
Down if c\colorFour{c}c is negative
Horizontal Translation
Right if d\colorThree{d}d is positive
Left if d\colorThree{d}d is negative

Watch Out!
When ddd is a negative value, we'll have two negative signs making the equation look more like this 
y=f(x−(−d))y=f(x+d)\begin{aligned}
y&=f(x - (-d)) \\
 y&= f(x + d)
\end{aligned}yy​=f(x−(−d))=f(x+d)​
Since ddd is negative, the graph will still move to the left.  This often feels opposite of our intuition since there is now a plus sign inside the function.  

Example 2
For each equation, describe the original function, and the type of translation.

1. y=x+3y = \sqrt{x} + 3y=x​+3

Function: f(x)=xf(x) = \sqrt{x}f(x)=x​
Translation:  Vertical translation up 3 units

2. y=1x+3y = \frac{1}{x + 3}y=x+31​

Function: f(x)=1xf(x) = \frac{1}{x}f(x)=x1​
Translation:  Horizontal translation left 3 units

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Example: Translations

At a particular factory shoe sizes for women are determined by measuring the length of the shoe in inches, and then putting this information into a function that returns the shoe size.  This function can be described by a table.  In general men's shoe will be labeled a size 1.5 less than the women's.   

1.  Let fff be the function that produces a women's shoe size given the shoe's length.  Write the equation that allows us to describe men's sizes using this function.

Let xxx be the length of a shoe.  
The equation  y=f(x)−1.5y = f(x) - 1.5y=f(x)−1.5  will take in an input of length, and produce an output of a men's sized shoe.  

2.  Use this equation to create an additional column giving us the Men's shoe sizes.  

3.  Describe how women's and men's shoe sizes are related to one another in terms of translations.

The function for men's shoe size is the same as the women's, but it has been translated down 1.5 sizes.  

Practice: Translations



The original function is f(x)=∣x∣f(x) = |x|f(x)=∣x∣
Fill in the blanks with the direction that the translated graph will go. 
 
1. y=∣x+3∣y = |x + 3|y=∣x+3∣
2.  y=∣x−3∣y = |x - 3|y=∣x−3∣
3.  y=∣x∣−3y = |x| - 3y=∣x∣−3
4.  y=∣x∣+3 y = |x| + 3y=∣x∣+3

1. Right, Left, Up, or Down

2. Right, Left, Up, or Down

3. Right, Left, Up, or Down

4. Right, Left, Up, or Down

I don't know

Practice: Translations
The original function and its graph are given.

f(x)=x2f(x) = x^2f(x)=x2

Match each translated equation with its new graph.

A.

B.

C.

D.

y=x2+2y = x^2 + 2y=x2+2

y=(x+2)2y = (x + 2)^2y=(x+2)2

y=(x−1)2y = (x - 1)^2y=(x−1)2

y=x2−1y = x^2 - 1y=x2−1

I don't know

Practice: Translations
The graph of a function f(x)f(x)f(x) is given as well as its translated graph.
Write the equation of the translated graph.

Enter your answer in the form

y=...

I don't know

Extra Practice

Vertical & Horizontal Translations

Practice: Vertical & Horizontal Translations
Let (a,b) (a, b)~(a,b) be some point on the function y=f(2x+4)+3y=f(2x+4)+3y=f(2x+4)+3. 

Vertical & Horizontal Translations

Practice: Vertical & Horizontal Translations
Let (3,−5) (3, -5)~(3,−5) be some point on the function y=−f(1−x)+3y=-f(1-x)+3y=−f(1−x)+3. 

Vertical & Horizontal Translations

Practice: Vertical & Horizontal Translations
Let f(x)=x2f(x)=x^2f(x)=x2. 
True or False:

Vertical & Horizontal Translations

Practice: Vertical & Horizontal Translations
Let f(x)=x3f(x)=x^3f(x)=x3. 
True or False:

Vertical & Horizontal Translations

Practice: Vertical & Horizontal Translations
The function y=f(x) y=f(x)~y=f(x) has been transformed to y=f(x−h)+ky=f(x-h)+ky=f(x−h)+k. Determine the values of h and k for each of the following transformations:

Vertical & Horizontal Translations

Practice: Vertical & Horizontal Translations
The function y=f(x) y=f(x)~y=f(x) has been transformed to y=f(x−h)+ky=f(x-h)+ky=f(x−h)+k. Determine the values of h and k for each of the following transformations: