Horizontal Shifts: Why Minus Moves a Graph Right

Add 3 outside a function and the graph rises 3. Subtract 3 inside and the graph moves right 3. Most people expect left, and the mismatch between what the sign looks like and what it does is the single most reliable source of lost marks in this topic.

The Rule

shifts the graph of right by units when is positive, and left when it is negative.

The trick that never fails: rewrite every plus as a minus. is , so and the shift is 4 left. Doing that on paper for a term or two makes the rule automatic.

Three Ways to See Why

Different explanations land for different people, so here are three.

1. The function is running late.

computes of a number three smaller than whatever you feed it. To get out of , you must put in . Every output happens three units later than it did for — so the graph is pushed forward, to the right.

2. Solve for where the interesting bit lands. If has a vertex where its input is 0, then has its vertex where , that is at . Setting the inside to zero always tells you where the key feature moved to, and it works even when there is a coefficient on .

3. Think of it as compensation. Subtracting 3 from the input weakens it, so must be 3 bigger to compensate. Bigger means further right.

Left and Right Together

Both directions are the same rule. The sign inside the bracket is the opposite of the direction of travel.

Every Point Moves Sideways Only

The point rule is as simple as it gets:

The -coordinate is never touched. That is why the shape, width and orientation of the graph are all preserved — a shifted parabola is the same parabola in a different place.

What Changes and What Doesn’t

The -intercept is the one that trips people. It does not “shift” — the line stays where it is, and the curve moving past it means a different point of the curve now sits there. Recompute it rather than sliding it.

When There Is a Coefficient on x

If has a coefficient inside the bracket, factor it out before reading the shift.

The shift is 3, not 6 — and there is also a horizontal compression by a factor of 2. Reading 6 straight off is the standard error here.

The safe method, again: set the inside to zero. gives .

Worked Example A: Read the Shift

Describe .

Right 8.

Worked Example B: A Plus Sign

Describe .

Rewrite as , so : left 2.

Worked Example C: On a Square Root

Describe against , and give its domain.

Right 5. The parent starts at , so this starts at .

Domain — shifted by 5. Range — unchanged, as always for a horizontal shift. See square root functions.

Worked Example D: A Coefficient Inside

Describe the horizontal shift in .

Factor: . The shift is right 4, together with a horizontal compression by a factor of 3.

Check by setting the inside to zero: gives . ✔

Worked Example E: An Asymptote Moves

Where is the vertical asymptote of ?

This is shifted right 6. The parent’s asymptote at moves to .

The horizontal asymptote stays at , because heights are unaffected.

Common Mistakes to Avoid

• Reading the sign as the direction. Minus means right.
• Not factoring out a coefficient. shifts by 3, not 6.
• Shifting the -intercept. Recompute it; it is not carried along.
• Changing the range. A horizontal shift never touches it.
• Thinking the shape stretches. Every point moves the same distance, so nothing distorts.
• Moving the horizontal asymptote. Only vertical ones move.
• Applying it to the output. is a vertical shift, a different transformation entirely.

Where Horizontal Shifts Lead Next

• The full picture. Function transformations puts this alongside the other three moves.
• Vertical shifts. Vertical shifts are the well-behaved counterpart — no sign surprises.
• Vertex form. The in is exactly this shift, which is why parabolas read the way they do.
• Horizontal compressions. squeezes by a factor of — backwards again, for the same input-side reason.
• Phase shift. In trigonometry the horizontal shift of a sine wave is called the phase shift, and follows this identical rule.

Practice Problems

Work each one before opening the answer.

Problem 1. Which way does move the graph?

Problem 2. Which way does move the graph?

Problem 3. Where is the vertex of ?

Problem 4. Where is the vertex of ?

Problem 5. Where does start?

Problem 6. Where does move under ?

Problem 7. What is the horizontal shift in ?

Problem 8. has domain . What is the domain of ?

Problem 9. has range . What is the range of ?

Problem 10. Where is the vertical asymptote of ?

Problem 11. Write the equation of shifted 6 left.

Problem 12. has -intercept . What is the -intercept of ?

Quick Reference

A horizontal shift is the counterintuitive member of the function transformations family; vertical shifts behave exactly as they read. The same appears in the vertex form of parabolas, absolute value functions and square root functions. More Algebra lessons are available.