Function Transformations: Shifts, Stretches, and Reflections

You do not need to learn a new graph for . It is the parabola you already know, moved and stretched. Transformations are the rules for that, and once you have them, one parent graph gives you an entire family.

The General Form

The Inside-Outside Rule

This one idea explains why half of transformations feel backwards.

Outside the function — , — changes the output, so it acts vertically and does exactly what it looks like. Add 2, the graph rises 2.

Inside the function — , — changes the input before ever sees it, so it acts horizontally and behaves in the opposite direction. Subtract 3, the graph moves right 3.

The reason is worth holding on to: has to wait until reaches 3 to produce what produced at 0. It is running late, so its graph is pushed forward. The full argument is in horizontal shifts.

A memory hook that survives exams: horizontal transformations lie, vertical ones tell the truth.

All Four on One Graph

Each transformation is independent. You can apply one, or all of them, and the underlying shape is never destroyed — a parabola stays a parabola, a V stays a V.

The Order Matters

Applying transformations in the wrong sequence gives the wrong graph.

Take and a point where .

• Stretch first, then shift: , then . ✔
• Shift first, then stretch: , then . ✘

The second is wrong because the stretch multiplied the as well. Order of operations decides it: in the multiplication happens before the addition, so the stretch must be applied before the vertical shift.

Tracking a Point

The fastest way to transform a graph is to transform a few of its points. For , a point on moves to

The -coordinate gains ; the -coordinate is multiplied by and then gains . Transform three or four key points and draw the curve through them.

Domain and Range

The pattern is the inside-outside rule again: horizontal changes touch the domain, vertical changes touch the range.

Worked Example A: Describe It

Describe .

Inside the bracket: means , so left 2. Outside: means down 5.

Worked Example B: With a Stretch

Describe .

Right 1, then stretched vertically by a factor of 3. No vertical shift.

Worked Example C: A Reflection

Describe .

Reflected across the -axis, then moved up 4. The reflection happens first — the is added after the negation.

Worked Example D: Building the Composite

Sketch where .

1. Right 3 gives , vertex .
2. Stretch by 2 and reflect, giving , vertex still but opening downward and twice as steep.
3. Up 3, so the vertex lands at .

Tracking a point confirms it: on moves to .

Worked Example E: Reading It Backwards

A parabola has been moved 4 left and 6 down from , and reflected across the -axis. Write its equation.

Left 4 means , so . Reflected means . Down 6 means .

Common Mistakes to Avoid

• Reading the horizontal shift the way it looks. moves right.
• Applying the vertical shift before the stretch. The stretch would multiply it.
• Thinking moves the graph. It scales and flips; it never shifts.
• Swapping which transformation hits the domain. Horizontal changes the domain, vertical changes the range.
• Assuming the shape changes. A parabola stays a parabola however you move it.
• Forgetting a reflection is a stretch by a negative. does both jobs at once.
• Multiplying by when tracking a point. It is , not .

Where Transformations Lead Next

• The individual moves. Horizontal shifts, vertical shifts and reflections each get a lesson of their own.
• Symmetry. A graph unchanged by a reflection has symmetry, which is transformations applied to a function’s own definition.
• Vertex form everywhere. Parabolas, absolute value functions and square root functions all use the same , , pattern.
• Horizontal stretches. compresses by a factor of — backwards again, for the same reason.
• Trigonometric graphs. Amplitude, period and phase shift are exactly these transformations under different names.

Practice Problems

Work each one before opening the answer.

Problem 1. Describe .

Problem 2. Describe .

Problem 3. Describe .

Problem 4. Describe .

Problem 5. Describe .

Problem 6. Describe .

Problem 7. Describe .

Problem 8. Where does move under ?

Problem 9. Where does move under ?

Problem 10. is moved right 2 and down 9. Write the equation.

Problem 11. In which order should be applied?

Problem 12. has domain and range . What are they for ?

Quick Reference

Transformations turn one parent graph into a whole family, which is why parabolas, absolute value functions and square root functions all share the same vertex form. The individual moves are covered in horizontal shifts, vertical shifts and reflections, and a graph that survives a reflection unchanged has symmetry. More Algebra lessons are available.