Translating a Graph

1 MIN READ · PRACTISE AT YOUR OWN PACE · FREE

Idea

A translation shifts a graph without changing its shape. Adding outside the function moves outputs vertically. Changing the input inside the function moves the graph horizontally.

Method

g(x)=f(x-h)+k.

A point $(u,f(u))$ becomes $(u+h,f(u)+k)$ . Positive $h$ shifts right, while positive $k$ shifts up.

Worked example

The vertex of $f(x)=x^2$ is $(0,0)$ . For $g(x)=(x-2)^2+3$ , that vertex moves to $(2,3)$ . The original points $(1,1)$ and $(-1,1)$ move to $(3,4)$ and $(1,4)$ , preserving symmetry about the new vertical line $x=2$ .

Common mistake

The sign inside looks reversed because the old input $0$ now occurs at $x=h$ . Derive the shift from that equation instead of memorizing a slogan.

Check your understanding

Where does the vertex move for $(x+4)^2-1$ ?

Show answer

To $(-4,-1)$ .

Explore

Move and scale a graph

Try this. Change one slider at a time. Positive h moves the vertex right; k moves it up; negative a reflects the curve and a = 0 makes it constant.

Vertical multiplier a1Horizontal shift h0Vertical shift k0

y = 1(x − (0))² + (0). Vertex (0, 0); opens up. Range y ≥ 0.

PAUSE & THINKA quick check, not a grade

Try it yourself.

Choose an answer and explain your reasoning to yourself. Use a hint if you get stuck.

How does y=f(x−3) compare with y=f(x)?

Shifted 3 units right

Shifted 3 units left

Shifted 3 units up

Hint 1 · Find a starting point

Follow an original input a to the new graph.

Hint 2 · Take the next step

The new input must satisfy x−3=a.

Show the reasoning

Answer: Shifted 3 units right

x=a+3 produces the old output f(a), so every point shifts right by 3.

Before moving on: what would make one of the other answers wrong? Saying why is part of understanding.

MAKE IT YOURS

Pause before the next idea.

Can you explain how the visualization connects to this lesson’s goal? If a step still feels uncertain, put this lesson on your review list and try the check again another day.

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