Vertical Scaling

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The following figure shows the graph of \(y = f\left( x \right)\) (the dotted curve) where f is some function, and the graph of \(y = g\left( x \right) = 2 f\left( x \right)\)(the solid curve):

 

Vertical Scaling - Graph 1

What was the effect of multiplying by 2? The graph has vertically scaled by a factor of 2. All values have doubled; if \(\left( {{x_0},{y_0}} \right)\)is a point on the graph of f, then \(\left({{x_0},2{y_0}} \right)\) is the corresponding point on the graph of g(note that the points where the graph crosses the x-axis, or the zeroes,are still the same – can you see why?).

The following figure shows the graph of \(y = f\left( x \right)\) (the dotted curve) where f is some function, and the graph of \(y = g\left( x \right) =  - \frac{1}{3}f\left( x\right)\) (the solid curve):

Vertical Scaling - Graph 2

What has happened? Not only has the graph scaled by a factor of 1/3, it has flipped as well, because of the negative sign of the multiplier.

In general, the graph of \(y = kf\left( x \right)\) can be obtained by vertically scaling the graph of \(y = f\left( x\right)\) by \(\left| k \right|\)units. If k is negative, the graph will also flip. If the magnitude of is greater than 1, then the graph will stretch vertically,because each value will get scaled by a factor of magnitude greater than 1. If the magnitude of k is less than 1, then the graph will compress vertically, because each value will reduce in magnitude.