Question

f(x) has three zeroes, r1, r2, and r3. If a vertical stretch is done to f(x), will the zeroes change? (please take in consideration all sort of graphs.)

Answer 1

Well if f(x) has three zeroes, I am assuming that you are referring to a polynomial, yes?

Answer 2

it can be a polynomial graph, or just three lines intersecting the x-axis. The graph isn't specified, but the question is really just asking about how the transformation will affect the zeroes.

Answer 3

Alright ill try my best to give a general solution then, in general a equation with three zeroes looks something like this:
\[f(x)=a(x-r_1)(x-r_2)(x-r_3)\]
We know that this value exists:
\[0=a(x-r_1)(x-r_2)(x-r_3)\]
because the solution would be \(x=r_1\) or \(x=r_2\) or \(x=r_3\).
So then watch very carefully:
Let us assume that a is a non-zero constant. If we divide both sides by a, we will find that:
\[0=a(x-r_1)(x-r_2)(x-r_3)=(x-r_1)(x-r_2)(x-r_3)\]
In general also:
Applying a vertical stretch which is our a-value to a function \(f(x)\) with three zeroes will not affect the zeroes:
\[0=a\times f(x)\rightarrow\frac{0}{a}=\frac{a\times f(x)}{a}\rightarrow0=f(x)\]
In other words, A function vertically stretched will equal zero when the function without the vertical stretch equals zero.

Answer 4

ah ok I get it. Thanks for the help, I was struggling with the theoretical stuff but this makes it a lot easier.