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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),~then y=f(x)+k y=f(x)+k~ gives a vertical translation by 'k' units.
  • If k>0,k>0, then the graph moves upwards
  • We say "A vertical translation of 'k' units upwards"
  • If 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. For a - d:
  1. Sketch the graphs all on the same grid
  2. Identify the value of 'k'
  3. g(x)=x2+1g(x)=x^2+1
  4. h(x)=x2+3h(x)=x^2+3
  5. m(x)=x2−2m(x)=x^2-2
  6. n(x)=x2−5n(x)=x^2-5
For a - d, the values of 'k' are:
  1. 1
  2. 3
  3. -2
  4. -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),~then y=f(x−h) y=f(x-h)~ gives a horizontal translation by 'h' units.
  • If h>0, h>0,~then the graph moves to the right
  • We say "A horizontal translation 'h' units right"
  • If 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.~For a - d:
  1. Sketch the graphs all on the same grid
  2. Identify the value of 'h'
  3. g(x)=(x+1)2g(x)=(x+1)^2
  4. h(x)=(x+3)2h(x)=(x+3)^2
  5. m(x)=(x−2)2m(x)=(x-2)^2
  6. n(x)=(x−5)2n(x)=(x-5)^2

For a - d, the values of 'h' are:
  1. -1
  2. -3
  3. 2
  4. 5
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Example: Vertical & Horizontal Translations

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

Part a)

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:

Sketch:

There is a vertical translation of 4 units up

Thus, k = 4

Part c)

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~, then identifying the following:
  1. The parent function
  2. The table of values for parent function
  3. The table of Values for transformed function
  4. Domain
  5. Range
The graph of the transformed function y=(x−3)2+1y=(x-3)^2+1 is:

Parent Function: y=x2y=x^2
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}

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}
Domain:{x∈R∣−∞<x<∞}\{x\in\mathbb{R}|-\infin<x<\infin\}
Range: {y∈R∣ −1≤y}\{y\in\mathbb{R}|~-1 \le y\}
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Example: Vertical & Horizontal Translations

Graph y=(x+1)3−1 y=(x+1)^3-1~, identifying the following:
  1. The parent function
  2. The table of values for parent function
  3. The table of Values for transformed function
  4. Domain
  5. Range
The graph of the transformed function y=(x+1)3−1y=(x+1)^3-1 is:

Parent Function: y=x3y=x^3
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}

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}
Domain:{x∈R∣−∞<x<∞}\{x\in\mathbb{R}|-\infin<x<\infin\}
Range: {y∈R∣ −∞<y<∞}\{y\in\mathbb{R}|~-\infin<y<\infin\}

Practice: Vertical & Horizontal Translations

The function y=f(x) y=f(x)~has been transformed to y=f(x−h)+ky=f(x-h)+k. Determine the values of h and k for each of the following transformations:
A.
h=−4 & k=7     ∴ y=f(x+4)+7h=-4~\&~k=7~~~~~\therefore~y=f(x+4)+7
B.
h=7 & k=−4     ∴ y=f(x−7)−4h=7~\&~k=-4~~~~~\therefore~y=f(x-7)-4
C.
h=−6     ∴ y=f(x+6)h=-6~~~~~\therefore~y=f(x+6)
D.
k=6    ∴ y=f(x)+6k=6~~~~\therefore~y=f(x)+6
6 units upward
6 units left
4 units downward and 7 units right
4 units left and 7 units upward

Practice: Vertical & Horizontal Translations

Let f(x)=xf(x)=\sqrt{x}.

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

Practice: Vertical & Horizontal Translations

Let (1,2) (1, 2)~be some point on the function y=f(x+3)−5y=f(x+3)-5.

Extra Practice
Vertical & Horizontal Translations
Practice: Vertical & Horizontal Translations
Let (a,b) (a, b)~be some point on the function y=f(2x+4)+3y=f(2x+4)+3.
Vertical & Horizontal Translations
Practice: Vertical & Horizontal Translations
Let (3,−5) (3, -5)~be some point on the function y=−f(1−x)+3y=-f(1-x)+3.
Vertical & Horizontal Translations
Practice: Vertical & Horizontal Translations
Let f(x)=x2f(x)=x^2.
True or False:
Vertical & Horizontal Translations
Practice: Vertical & Horizontal Translations
Let f(x)=x3f(x)=x^3.
True or False:
Vertical & Horizontal Translations
Practice: Vertical & Horizontal Translations
The function y=f(x) y=f(x)~has been transformed to y=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)~has been transformed to y=f(x−h)+ky=f(x-h)+k. Determine the values of h and k for each of the following transformations: