Question

Does anyone know the answer to this question?

Is the graph of y = f (a x) ever the same as the graph of y = a f(x) for linear functions? If so, when?

Answer 1

Consider the template:

f(x) = mx+b

• m = slope
• b = y intercept

Then we replace x with ax to get

f(ax) = m(ax) + b

f(ax) = a(mx) + b

Set this equal to a*f(x) = a*(mx+b) to find that:

a(mx) + b = a*(mx+b)

amx + b = amx + ab

b = ab

b-ab = 0

b(1-a) = 0

b = 0 or 1-a = 0

b = 0 or a = 1

If a = 1, then f(ax) = f(1x) = f(x), and also a*f(x) = 1*f(x) = f(x)

Nothing really special happens when a = 1 since we basically collapse back to the original function.

If b = 0, then f(ax) = a*f(x) is the case. This is when the graph passes through the origin.

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In summary: The graphs are the same when a = 1 or b = 0

• When a = 1, the equations don't change since we're multiplying by 1.
• When b = 0, the equations don't change because we're basically comparing m(ax) to a(mx). They are the same regardless of the order of the variables.

For situations where you don't have control of the value of b, then a = 1 is only thing you can rely on.