Answer:
Stretch in any direction (horizontal or vertical) means the corresponding coordinates are multiplied by the stretch factor: a horizontal stretch multiplies the x-coordinates by the stretch factor; a vertical stretch multiplies the y-coordinates by the stretch factor.
Step-by-step explanation:
If you know the coordinates, you can apply the stretch factor directly to the coordinates.
For example, consider the point (5, 25).
A horizontal stretch (only) by a factor of 3 will move this point to (15, 25).
A vertical stretch (only) by a factor of 3 will move the original point to (5, 75).
Note that only the corresponding coordinate is multiplied by 3.
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The confusion can arise when this stretch concept is applied to the transformation of a function.
Consider the function f(x) = x^2. The point used in the example above is ...
(5, f(5))
Vertical Stretch Function Transformation
If we want to transform the function to one that is vertically stretched by a factor of 3, we can simply multiply the function value by 3:
g(x) = 3·f(x)
Then ...
(5, g(5)) = (5, 75) . . . . . . the location of the vertically stretched point in the above example.
Horizontal Stretch Function Transformation
If we want to stretch the above f(x) function horizontally by a factor of 3, we want a h(x) function that will produce the point (15, h(15)) = (15, 25). We can get that using f(x), but the argument to f(x) for that y-coordinate must be 5, not 15. This means the transformation must be ...
h(x) = f(x/3)
Dividing the function argument by the stretch factor means the argument must be larger by that factor in order to give the same function result.
(15, 25) = (15, h(15)) = (15, f(15/3)) = (15, f(5))
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Summary
Vertical stretch of a coordinate by the factor "a": (x, y) ⇒ (x, ay)
Vertical stretch of a function by the factor "a": g(x) = a·f(x)
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Horizontal stretch of a coordinate by the factor "a": (x, y) ⇒ (ax, y)
Horizontal stretch of a function by the factor "a":
(x, y) ⇒ (ax, h(ax)), where h(x) = f(x/a), so h(ax) = f(ax/a) = f(x)
