AP PRECALCULUS — UNIT 1B · POLYNOMIAL & RATIONAL FUNCTIONS
1.12A — Translations of Functions
Notes — Shifting Graphs Vertically and Horizontally
💡 Learning Objectives
By the end of this lesson you will be able to (AP CED 1.12.A):
- Apply vertical and horizontal translations to the graph of a function.
- Write the formula for a translated function given the translation.
- Describe the effect of a translation on key features (zeros, asymptotes, domain, range).
1. Vertical Translations
Adding a constant k to the OUTPUT of a function shifts its graph vertically. If k > 0, the graph moves up; if k < 0, the graph moves down. The shape does not change — every point moves by the same vertical distance.
💡 Definitions
Vertical Translation.
- The graph of g(x) = f(x) + k is the graph of f(x) translated k units up (if k > 0) or |k| units down (if k < 0).
- The point (x, y) on the graph of f becomes (x, y + k) on the graph of g.
2. Horizontal Translations
Replacing the INPUT x with (x − h) shifts the graph horizontally. Pay close attention to the sign — a common point of confusion. Replacing x with (x − 2) shifts the graph RIGHT by 2; replacing x with (x + 2) — which is (x − (−2)) — shifts LEFT by 2.
💡 Definitions
Horizontal Translation.
- The graph of g(x) = f(x − h) is the graph of f(x) translated h units to the right (if h > 0) or |h| units to the left (if h < 0).
- The point (x, y) on the graph of f becomes (x + h, y) on the graph of g.
Left: vertical translations of f(x) = x². Right: horizontal translations. Notice horizontal shifts are opposite to what the sign suggests — (x − 2)² shifts RIGHT, not left.
⚠️ Common Mistake
The sign on horizontal translations fools everyone at first. g(x) = f(x − 3) shifts RIGHT by 3, because you need a larger input x to reach the same output as before. If this seems backwards, test with a specific point: f(0) = f(0 − 0), but g(3) = f(3 − 3) = f(0). The same output now appears at input 3 instead of 0 — so the graph moved RIGHT.
💡 Quick Check
Describe each transformation of f(x) in words.
- g(x) = f(x) − 5 → shift DOWN 5 units
- g(x) = f(x + 4) → shift LEFT 4 units (because it's f(x − (−4)))
- g(x) = f(x − 2) + 7 → shift RIGHT 2 and UP 7
3. Combined Translations
Vertical and horizontal translations can be combined. The form g(x) = f(x − h) + k represents a shift of h units horizontally and k units vertically. Order does not matter — translations commute.
📘 Example
Example 1 — Writing a translated function
Let f(x) = |x|. Write the equation of the function g obtained by shifting f three units right and two units down.
Horizontal: replace x with (x − 3). Vertical: subtract 2.
g(x) = |x − 3| − 2.
📘 Example
Example 2 — Reading the translation from the formula
Describe the transformation that takes y = x² to g(x) = (x + 1)² − 4.
(x + 1) = (x − (−1)) → shift LEFT 1 unit. −4 at the end → shift DOWN 4.
So g is the parabola y = x² shifted 1 left and 4 down. New vertex: (−1, −4).
4. Effect on Key Features
- Vertical translations move zeros UP or DOWN — potentially changing the number of zeros.
- Horizontal translations move zeros LEFT or RIGHT by the same shift.
- Asymptotes also shift: a horizontal asymptote y = c becomes y = c + k under a vertical shift; a vertical asymptote x = a becomes x = a + h under a horizontal shift.
- Domain shifts horizontally; range shifts vertically.
🎯 AP Tip
On the AP Exam, you may be asked to describe a transformation in words AND to write the new formula. Lead with the word description ('f shifted 3 right and 2 up'), then write the formula g(x) = f(x − 3) + 2. Both components are required for full credit.
📘 Try This
Let f(x) = 1/x. Write g(x) for each translation.
- Right 3, up 1: g(x) = 1/(x − 3) + 1
- Left 2, down 4: g(x) = 1/(x + 2) − 4
- State the new vertical and horizontal asymptotes of each g.
5. Summary
- Vertical translation: g(x) = f(x) + k. Up if k > 0, down if k < 0.
- Horizontal translation: g(x) = f(x − h). Right if h > 0, left if h < 0.
- Combined: g(x) = f(x − h) + k. Order doesn't matter.
- All key features shift along with the graph — zeros, asymptotes, extrema, domain, range.