Wize High School Grade 12 Pre-Calculus Textbook > Transformations

Vertical & Horizontal Translations

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Vertical Translations

A vertical translation moves every point on some function by a fixed distance vertically. 

If y=f(x), y=f(x),~y=f(x), then y=f(x)+k y=f(x)+k~y=f(x)+k  gives a vertical translation by 'k' units.
If k>0,k>0,k>0, then the graph moves upwards
We say "A vertical translation of 'k' units upwards"
If k<0,k<0,k<0, then the graph moves downwards
We say "A vertical translation of 'k' units downwards"

Example 

Let f(x)=x2.f(x)=x^2. f(x)=x2. For a - d:
Sketch the graphs all on the same grid
Identify the value of 'k'
g(x)=x2+1g(x)=x^2+1g(x)=x2+1
h(x)=x2+3h(x)=x^2+3h(x)=x2+3
m(x)=x2−2m(x)=x^2-2m(x)=x2−2
n(x)=x2−5n(x)=x^2-5n(x)=x2−5

For a - d, the values of 'k' are:
1
3
-2
-5

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Horizontal Translations
A horizontal translation moves every point on some function by a fixed distance horizontally

If y=f(x), y=f(x),~y=f(x), then y=f(x−h) y=f(x-h)~y=f(x−h)  gives a horizontal translation by 'h' units.
If h>0, h>0,~h>0, then the graph moves to the right
We say "A horizontal translation 'h' units right"
If h<0, h<0,~h<0, then the graph moves to the left
We say "A horizontal translation 'h' units left"

Example 

Let f(x)=x2. f(x)=x^2.~f(x)=x2. For a - d:
Sketch the graphs all on the same grid
Identify the value of 'h'
g(x)=(x+1)2g(x)=(x+1)^2g(x)=(x+1)2
h(x)=(x+3)2h(x)=(x+3)^2h(x)=(x+3)2
m(x)=(x−2)2m(x)=(x-2)^2m(x)=(x−2)2
n(x)=(x−5)2n(x)=(x-5)^2n(x)=(x−5)2

For a - d, the values of 'h' are:
-1
-3
2
5

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Example: Vertical & Horizontal Translations
Sketch the transformed graph of the function, f(x)f(x)f(x) (graphed below), stating and identifying the values of 'h' and 'k':
f(x−3)f(x-3)f(x−3)
f(x)+4f(x)+4f(x)+4
f(x+1)−2f(x+1)-2f(x+1)−2

Part a)

f(x−3)f(x-3)f(x−3)

Sketch:

There is a horizontal translation 3 units right

Thus, h = 3

Part b)

f(x)+4:f(x)+4:f(x)+4:

Sketch:

There is a vertical translation of 4 units up

Thus, k = 4

Part c)

f(x+1)−2:f(x+1)-2:f(x+1)−2:

Sketch:

There is a vertical translation 2 units down and a horizontal translation 1 unit left.

Thus, h = -1 & k = -2

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Example: Vertical & Horizontal Translations
Graph y=(x−3)2+1 y=(x-3)^2+1~y=(x−3)2+1 , then identifying the following: 
The parent function
The table of values for parent function
The table of Values for transformed function
Domain 
Range

The graph of the transformed function y=(x−3)2+1y=(x-3)^2+1y=(x−3)2+1 is: 

Parent Function: y=x2y=x^2y=x2
Table of Values for Parent Function: 
xy−24−11001124\begin{array}
{|c|c|}
\hline
x&y\\\hline
-2&4\\
-1&1\\
0&0\\
1&1\\
2&4\\\hline
\end{array}x−2−1012​y41014​​

Table of Values for Transformed Function: 
xy−226−1170101522\begin{array}
{|c|c|}
\hline
x&y\\\hline

-2&26\\
-1&17\\
0&10\\
1&5\\
2&2\\
\hline
\end{array}x−2−1012​y26171052​​
Domain:{x∈R∣−∞<x<∞}\{x\in\mathbb{R}|-\infin<x<\infin\}{x∈R∣−∞<x<∞} 
Range:  {y∈R∣ −1≤y}\{y\in\mathbb{R}|~-1 \le y\}{y∈R∣ −1≤y}

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Example: Vertical & Horizontal Translations
Graph y=(x+1)3−1 y=(x+1)^3-1~y=(x+1)3−1 , identifying the following: 
The parent function
The table of values for parent function
The table of Values for transformed function
Domain 
Range

The graph of the transformed function y=(x+1)3−1y=(x+1)^3-1y=(x+1)3−1 is:

Parent Function: y=x3y=x^3y=x3
Table of Values for Parent Function: 
xy−2−8−1−1001128\begin{array}
{|c|c|}
\hline
x&y\\\hline
-2&-8\\
-1&-1\\
0&0\\
1&1\\
2&8\\\hline
\end{array}x−2−1012​y−8−1018​​

Table of Values for Transformed Function: 
xy−3−9−2−2−1−10−017\begin{array}
{|c|c|}
\hline
x&y\\\hline
-3&-9\\
-2&-2\\
-1&-1\\
0&-0\\
1&7\\\hline
\end{array}x−3−2−101​y−9−2−1−07​​
Domain:{x∈R∣−∞<x<∞}\{x\in\mathbb{R}|-\infin<x<\infin\}{x∈R∣−∞<x<∞} 
Range:  {y∈R∣ −∞<y<∞}\{y\in\mathbb{R}|~-\infin<y<\infin\}{y∈R∣ −∞<y<∞}

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:

A.

h=7 & k=−4     ∴ y=f(x−7)−4h=7~\&~k=-4~~~~~\therefore~y=f(x-7)-4h=7 & k=−4     ∴ y=f(x−7)−4

B.

k=6    ∴ y=f(x)+6k=6~~~~\therefore~y=f(x)+6k=6    ∴ y=f(x)+6

C.

h=−4 & k=7     ∴ y=f(x+4)+7h=-4~\&~k=7~~~~~\therefore~y=f(x+4)+7h=−4 & k=7     ∴ y=f(x+4)+7

D.

h=−6     ∴ y=f(x+6)h=-6~~~~~\therefore~y=f(x+6)h=−6     ∴ y=f(x+6)

6 units upward

6 units left

4 units downward and 7 units right

4 units left and 7 units upward

I don't know

Practice: Vertical & Horizontal Translations
Let f(x)=xf(x)=\sqrt{x}f(x)=x​. 

True or False:
The graph of the transformed function f(x)=x−3f(x)=\sqrt{x-3}f(x)=x−3​ will affect the range of the function.

Answer

I don't know

Practice: Vertical & Horizontal Translations
Let (1,2) (1, 2)~(1,2) be some point on the function y=f(x+3)−5y=f(x+3)-5y=f(x+3)−5. 

A) What point must be on the graph

y = f(x)

B) What point must be on the graph

y = f(x - 1) + 1

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: