How horizontal shift works in trig graph transformations

Key concept: When a trigonometric function is shifted, the tricky part is that the motion happens inside the parentheses, not outside. That’s why the sign feels backwards at first.

Step by step

For a function written in the transformed form

$y=A\sin(B(x-h))+D \quad \text{or} \quad y=A\cos(B(x-h))+D,$

the horizontal shift is determined by the value of $h$.

Step 1: Read the inside carefully

• If $h>0$, the graph shifts right by $h$ units
• If $h<0$, the graph shifts left by $|h|$ units

Step 2: Match the form

Notice the expression is usually written as $(x-h)$, not $(x+h)$. That means the sign appears reversed compared with the motion.

Step 3: Example

• $y=\sin(x-3)$ shifts right 3 units
• $y=\sin(x+3)=\sin(x-(-3))$ shifts left 3 units

So the statement is correct: a positive $h$ means a shift to the right, and a negative $h$ means a shift to the left.

Pitfall alert

Students often read the sign literally and say $(x+2)$ means right 2. It does the opposite. The safest habit is to rewrite the inside as $x-h$ first, then identify $h$. That removes most sign mistakes on graph transformations.

Try different conditions

If the function is written as $y=f(x+h)$ instead of $y=f(x-h)$, the shift is still the same idea, but the visible sign flips. For example, $f(x+4)$ means the graph moves left 4 units. If there is also a factor like $B(x-h)$, first handle the horizontal stretch by $\frac{1}{|B|}$, then read the shift $h$.

Further reading

phase shift, horizontal translation, graph transformation