Stretching a Graph Vertically

Multiply every height

Start from $f\left(x\right)=x^2$f(x) = x² and multiply the whole function by 2: y = 2f(x), which is $y=2x^2$y = 2x². The squaring happens first and the multiplying after it, so the 2 acts on the output. At x = 1 the height goes from 1 to 2. At x = 2 it goes from 4 to 8, and at x = 3 from 9 to 18. Every height is doubled, and every point stays above the same x.

So each point (x, y) of $y=x^2$y = x² moves to (x, 2y), straight up or down, never sideways. The curve is pulled away from the x-axis and looks steeper. This is called a vertical stretch with scale factor 2, or a stretch parallel to the y-axis. In general, y = af(x) moves each point (p, q) of y = f(x) to (p, aq).

The white curve is $y=x^2$y = x² and the gold curve is $y=2x^2$y = 2x². Above x = 2, the point (2, 4) has moved straight up to (2, 8).

Multiplying is not adding

Compare $y=2x^2$y = 2x² with $y=x^2+2$y = x² + 2. Adding 2 moves every point up by the same amount, 2, which is a translation: at x = 1 the height goes from 1 to 3, and at x = 3 from 9 to 11. Multiplying by 2 moves each point by an amount that depends on its height: at x = 1 the height goes from 1 to 2, a move of 1, and at x = 3 from 9 to 18, a move of 9.

So a stretch moves points far from the x-axis a long way and points near it hardly at all. That is why the shape changes, and not only the position.

A factor between 0 and 1

The same rule works when the factor is less than 1. For $y=\frac{1}{2}{x}^2$y = ½x², every height is halved: (2, 4) moves to (2, 2), and (3, 9) moves to (3, 4.5). The curve is pressed toward the x-axis and looks flatter.

This is still called a stretch, with scale factor ½. A scale factor greater than 1 pulls the graph away from the x-axis; a scale factor between 0 and 1 pushes it toward the axis.

The white curve is $y=x^2$y = x² and the gold curve is $y=\frac{1}{2}{x}^2$y = ½x². Above x = 2, the height 4 has been halved to 2.

A negative factor

For $y=-x^2$y = −x², the factor is −1. Every height changes sign: (2, 4) moves to (2, −4), and (3, 9) moves to (3, −9). Each point goes to the same distance on the other side of the x-axis, so the graph is reflected in the x-axis, and the parabola now opens downward.

A factor such as −2 does both things at once: $y=-2x^2$y = −2x² doubles every height and reflects it, so (2, 4) moves to (2, −8). Because every height changes sign, the highest point of a graph becomes the lowest point of the new one. If the greatest value of f(x) is 7, the least value of −f(x) is −7.

The white curve is $y=x^2$y = x² and the gold curve is $y=-x^2$y = −x², its reflection in the x-axis. The point (2, 4) has moved to (2, −4).

Points on the x-axis stay where they are

A point on the x-axis has height 0, and a × 0 = 0 for every a. So a vertical stretch leaves every point on the x-axis exactly where it is: the graph of y = af(x) crosses the x-axis at the same places as y = f(x).

Take $f\left(x\right)=x^2-4$f(x) = x² − 4, which crosses the x-axis at x = −2 and x = 2. The graph of $y=2f\left(x\right)=2x^2-8$y = 2f(x) = 2x² − 8 crosses at the same two places: at x = 2, $2\times 2^2-8=8-8=0$2 × 2² − 8 = 8 − 8 = 0. The vertex, which is 4 below the axis at (0, −4), moves to twice that depth, (0, −8). The points on the axis stay fixed while the rest of the curve moves away from it.

The white curve is $y=x^2-4$y = x² − 4 and the gold curve is $y=2\left(x^2-4\right)$y = 2(x² − 4). Both cross the x-axis at (−2, 0) and (2, 0). The vertex has moved from (0, −4) down to (0, −8).

A straight line stretched

The rule works for every graph. Stretch the line y = x + 1 with scale factor 2: y = 2(x + 1) = 2x + 2. The line crosses the x-axis at x = −1, and so does the new one: 2 × (−1) + 2 = 0. Its y-intercept doubles from 1 to 2, and its gradient doubles from 1 to 2, because every rise is twice as tall over the same run.

So the two lines meet on the x-axis, at (−1, 0), and the stretched line is steeper. Doubling the heights of a straight line doubles its gradient.

The white line is y = x + 1 and the gold line is y = 2x + 2, its vertical stretch with scale factor 2. They cross the x-axis at the same point, (−1, 0).

The usual mistakes

Stretching the input instead of the output. If (3, 5) is on y = f(x), then on y = 2f(x) it moves to (3, 10). The x stays the same; (6, 5) would be a sideways stretch.

Adding instead of multiplying. y = 2f(x) doubles the heights: (3, 5) goes to (3, 10), not (3, 7). Adding 2 is a different transformation, a move up.

Moving the crossings. If y = f(x) crosses the x-axis at x = 4, then y = 2f(x) crosses there too, because 2 × 0 = 0. It does not move to x = 8.

Forgetting the reflection. A negative factor turns the graph over: the least value of −3f(x) comes from the greatest value of f(x).