HORIZONTAL TRANSFORMATIONS OF FUNCTIONS
y = cos 3(x - 60) versus y = cos (3x - 60)
| QUESTION A: How to draw the graph y = cos 3(x - 60) from y = cos x |
QUESTION B: How to draw the graph y = cos (3x - 60) from y = cos x |
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| SOLUTION A | SOLUTION B1 | |
| Replace (x) by (3x) in
y = cos x (Compress the graph horizontally by a factor of 1/3) |
Replace (x) by (x - 60) in
y = cos x (Translate the graph 60o right) |
|
| y = cos x will become y = cos 3x | y = cos x will become y = cos (x - 60) | |
| Replace (x) by (x - 60) in
y = cos 3x (Translate the graph 60o right) |
Replace x by 3x in
y = cos (x - 60) (Compress the graph horizontally by a factor of 1/3) |
|
| y = cos 3x will become y = cos 3(x - 60) | y = cos (x - 60) will become y = cos (3x - 60) | |
| SOLUTION B2 (Second Method) | ||
| cos (3x - 60) = cos 3(x - 20) Draw
y = cos 3(x - 20) by using same method given at SOLUTION A. |
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x |
-9 |
-8 |
-7 |
-6 |
-5 |
-4 |
-3 |
-2 |
-1 |
0 |
1 |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
9 |
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f (x) |
-2 |
-1 |
0 |
1 |
2 |
3 |
2.5 |
2 |
1.5 |
1 |
1.5 |
2 |
2.5 |
3 |
3.5 |
4 |
3 |
2 |
1 |
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f (x+1) |
-1 |
0 |
1 |
2 |
3 |
2.5 |
2 |
1.5 |
1 |
1.5 |
2 |
2.5 |
3 |
3.5 |
4 |
3 |
2 |
1 |
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x |
-9 |
-8 |
-7 |
-6 |
-5 |
-4 |
-3 |
-2 |
-1 |
0 |
1 |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
9 |
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f (x) |
-2 |
-1 |
0 |
1 |
2 |
3 |
2.5 |
2 |
1.5 |
1 |
1.5 |
2 |
2.5 |
3 |
3.5 |
4 |
3 |
2 |
1 |
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f (x-2) |
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-2 |
-1 |
0 |
1 |
2 |
3 |
2.5 |
2 |
1.5 |
1 |
1.5 |
2 |
2.5 |
3 |
3.5 |
4 |
3 |
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x |
-9 |
-8 |
-7 |
-6 |
-5 |
-4 |
-3 |
-2 |
-1 |
0 |
1 |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
9 |
|
f (x) |
-2 |
-1 |
0 |
1 |
2 |
3 |
2.5 |
2 |
1.5 |
1 |
1.5 |
2 |
2.5 |
3 |
3.5 |
4 |
3 |
2 |
1 |
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f (x/2) |
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3 |
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2.5 |
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2 |
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1.5 |
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1 |
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1.5 |
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2 |
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2.5 |
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3 |
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x |
-9 |
-8 |
-7 |
-6 |
-5 |
-4 |
-3 |
-2 |
-1 |
0 |
1 |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
9 |
|
f (x) |
-2 |
-1 |
0 |
1 |
2 |
3 |
2.5 |
2 |
1.5 |
1 |
1.5 |
2 |
2.5 |
3 |
3.5 |
4 |
3 |
2 |
1 |
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f (2x) |
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-1 |
1 |
3 |
2 |
1 |
2 |
3 |
4 |
2 |
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Horizontal Transformation: Replace x by (x + 1): Transfer the graph horizontally 1 unit to the left
Horizontal Transformation: Replace x by (x - 2): Transfer the graph horizontally 2 unit to the right
Horizontal Expansion: Replace x by x/2: Expand the graph horizontally by a factor of 2
Horizontal Compression: Replace x by 2x: Compress the graph horizontally by a factor of 1/2