Vertical Shifts: Moving a Graph Up and Down

Vertical shifts are the transformation that behaves. There is no sign trick, no factoring, no compensating — add 3 to the output and the graph rises 3. It is worth understanding why it is so well behaved, because that reason is exactly what makes horizontal shifts misbehave.

The Rule

moves the graph up by when is positive, and down when it is negative.

That really is all of it. The reason there is no surprise here is that is added after has done its work. The function computes its output, then 3 is added — so the output is genuinely 3 bigger, and the point sits 3 higher.

Compare , where the subtraction happens to the input, before the function runs, and has to be compensated for. That is the whole difference, and it is covered in horizontal shifts.

Up and Down

The point rule is the mirror image of the horizontal one:

The -coordinate is untouched. Width, orientation and curvature are all preserved.

What Changes and What Doesn’t

Two of these are worth dwelling on.

The -intercept shifts cleanly. It is an output, and every output moves by . This is the opposite of a horizontal shift, where the -intercept has to be recomputed from scratch.

The -intercepts do not simply shift. They are where the graph meets a fixed line, and the graph is moving relative to that line. Lift by 3 and the two intercepts move closer together; lift it by 4 and they merge into one; lift it by 5 and they disappear. The count itself can change, which never happens under a horizontal shift.

Horizontal Versus Vertical

and look almost identical on the page and do completely different things. Check whether the number is inside or outside the bracket before anything else.

Asymptotes Rise Too

is the reciprocal curve lifted 2. Its horizontal asymptote moves from to ; its vertical asymptote stays at , because no -value has changed. This is how rational functions end up with asymptotes at heights other than zero.

Worked Example A: Read the Shift

Describe .

Down 11.

Worked Example B: The Range Moves

has range . What is the range of ?

. The domain is unchanged, whatever it was.

Worked Example C: Intercepts Disappearing

How many -intercepts does have, and ?

has two, at .

Adding 12 gives , whose vertex is at — above the axis, so none. The shift lifted the curve clear of the -axis.

Worked Example D: On a Square Root

Describe and give its domain and range.

Down 4. It starts at .

Domain — unchanged. Range — shifted down 4.

Worked Example E: Both Shifts at Once

Describe , and say where ends up.

Right 2 and up 5.

The point rule combines: .

Common Mistakes to Avoid

• Confusing with . Outside is vertical, inside is horizontal.
• Shifting the domain. A vertical shift never touches it.
• Sliding the -intercepts by . They move relative to a fixed axis, and can vanish entirely.
• Leaving a horizontal asymptote behind. It rises with the graph.
• Moving the vertical asymptote. It does not move at all.
• Expecting a sign trick. There isn’t one here; plus means up.
• Applying the shift before a stretch. In the stretch comes first — see function transformations.

Where Vertical Shifts Lead Next

• The full set. Function transformations covers how this combines with the other three.
• Vertex form. The in is exactly this shift — see parabolas.
• Reflections. Reflections are what happens when goes negative rather than changing.
• Solving by shifting. “How far must this curve move to touch the axis?” is a vertical-shift question in disguise.
• Baseline modelling. A fixed fee added to a variable cost is a vertical shift of the cost graph.

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 does move under ?

Problem 5. has domain . What is the domain of ?

Problem 6. has range . What is the range of ?

Problem 7. How many -intercepts does have?

Problem 8. What is the horizontal asymptote of ?

Problem 9. What is the vertical asymptote of ?

Problem 10. Is the same as ?

Problem 11. Write the equation of moved down 6.

Problem 12. has two -intercepts. What vertical shift leaves it with exactly one?

Quick Reference

Vertical shifts are the straightforward half of function transformations, and their good behaviour is exactly what horizontal shifts lack. The is the same one in the vertex form of parabolas, and lifting a rational function is how its horizontal asymptote ends up somewhere other than zero. More Algebra lessons are available.