Function Transformations • Topic 1 of 4

Translations

A translation slides a graph without changing its shape. Adding a constant outside the function shifts it vertically: f(x) + k moves up by k, f(x) − k moves down. Changing the input shifts it horizontally, and the direction is counter-intuitive: f(x − h) moves right by h, while f(x + h) moves left. So starting from y = x², the graph of y = (x − 3)² is shifted right 3, and y = x² + 4 is shifted up 4. Vertical shifts match their sign; horizontal shifts are opposite. The SAT tests writing the new equation after a described shift, so watch the “inside is opposite” rule.

sat33t1 graphOTranslationy = (x − 3)² + 2: right 3, up 2y = x² (parent)transformed

✅ Solved examples

1. Shift y = x² up 5. New equation?
Add 5 outside: y = x² + 5.
2. Shift y = x² right 3. New equation?
Replace x with x − 3: y = (x − 3)².
3. Shift y = x² left 2. New equation?
Replace x with x + 2: y = (x + 2)².
4. Shift y = √x down 4. New equation?
Subtract 4 outside: y = √x − 4.

✏️ Practice — try these, take hints as needed

1. Shift y = x² down 6. New equation?
Subtract outside.
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y = x² − 6.
2. Shift y = x² left 5. New equation?
x → x + 5.
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y = (x + 5)².
3. Shift y = x² right 7. New equation?
x → x − 7.
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y = (x − 7)².
4. Shift y = |x| up 3. New equation?
Add outside.
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y = |x| + 3.
5. Shift y = √x right 2. New equation?
x → x − 2.
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y = √(x − 2).

📝 Topic test — 8 questions

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