Stretching and Compressing a Graph

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Idea

Scaling changes distances from an axis. Vertical multiplication changes output distances; input multiplication changes the horizontal distance needed to reach an old input.

Method

For $g(x)=a f(bx)$ with nonzero $a,b$ , $(u,v)$ becomes $(u/b,av)$ . Thus $|a|>1$ stretches vertically, while $|b|>1$ compresses horizontally. Negative factors also reflect.

Worked example

For $f(x)=x^2$ , $g(x)=3f(x)$ triples every output. The point $(2,4)$ becomes $(2,12)$ . For $h(x)=f(2x)$ , the same old input $2$ is reached at $x=1$ , so its point becomes $(1,4)$ . Although $f(2x)=4x^2$ here, horizontal and vertical scaling are different constructions.

Common mistake

A multiplier of zero collapses a graph and is not an invertible stretch. Track this as its own case.

Check your understanding

Where does $(4,3)$ move under $g(x)=2f(2x)$ ?

Show answer

To $(2,6)$ .

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.

A point (2,3) lies on y=f(x). Which corresponding point lies on y=2f(x)?

(2,6)

(4,3)

(2,5)

Hint 1 · Find a starting point

The factor is outside the function.

Hint 2 · Take the next step

It multiplies outputs while keeping inputs fixed.

Show the reasoning

Answer: (2,6)

The height 3 doubles to 6 at the same x=2.

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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