Shifting a Graph

1 MIN READ · PRACTISE AT YOUR OWN PACE · FREE

Idea

Shifts let you reuse a familiar graph while carrying its domain and characteristic points along with it. This matters in calculus because singularities and limits move too.

Method

For $g(x)=f(x-h)+k$ , translate every point by $(h,k)$ . A forbidden input $u$ of $f$ becomes the forbidden input $u+h$ of $g$ . Vertical shifts do not change the domain.

Worked example

Starting from $f(x)=1/x$ , the function $g(x)=1/(x-2)+3$ has vertical asymptote $x=2$ and horizontal asymptote $y=3$ . Its domain excludes $2$ and its range excludes $3$ . The point $(1,1)$ becomes $(3,4)$ .

Common mistake

A shift inside a denominator is still an input shift. Solve the denominator-zero equation rather than inferring a sign visually.

Check your understanding

Find the domain and starting point of $\sqrt{x+4}-2$ .

Show answer

Domain $[-4,\infty)$ ; starting point $(-4,-2)$ .

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 is y=f(x)+4 shifted from y=f(x)?

4 units up

4 units right

4 units down

Hint 1 · Find a starting point

The addition happens after f is evaluated.

Hint 2 · Take the next step

It changes the output, not the input.

Show the reasoning

Answer: 4 units up

Each point (x,y) becomes (x,y+4).

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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