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Graphs can be shifted and reflected, stretched and squashed. These are all known as transformations. And so now we’re going to look at stretching and squashing. We will discover how the equation of the graph looks, compared to its changed shape. Let’s start with vertical stretches and squashes as they’re a little easier.
As with all vertical transformations, we apply the transformation to the whole function - so the outside.
Vertical Transformations:
shifts y = f(x) + a the curve up / down
y = − f(x) reflects the curve in the x-axis
y = af(x) stretches/squashes the curve vertically
See how the number goes here?
So, because this curve has the equation, y = 2 x-squared, the 2 means that we need to double every y value.
So here, y was at 1 which needs to double to 2.
Here 4 needs to double to 8, So every y-coordinate double in size, If the new curve was y = 3 x-squared, then every y coordinate would need to multiply by 3. So, 1 goes to 3 and so on.
See what happens when the equation is y = ½ x-squared... The y coordinates half in size… So, 4 goes to 2. If you have to transform a graph yourself, just take it point by point. So, we have the graph of y = f(x) and we need y = 3 f(x).
So, we divide each y-coordinate by 3 Negative 9 moves to negative 3, Negative 6 goes to negative 2, 3 goes to 1. And you’ll end up with your transformed graph… so y = ⅓ f(x) would be squashed vertically by a third. Horizontal stretches and squashes aren’t much different. As with all horizontal transformations, we apply the transformation directly to the x’s. See how the ‘2’ just goes with the x and ignores the 8. Notice how ‘2’ seems to squash the curve horizontally, whereas ‘½’ stretches the curve.
As with all horizontal transformations they’re a little strange. So, any numbers bigger than 1 will squash the curve, and any numbers smaller than 1 will stretch the curve horizontally.
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