Question

the graph of a function f(x) passes through the following points: (0, 0), (1, -2), (2, 0) which of the following could be f(x) f(x)= 2x^2-4x f(x)= -2x^2f(x)= -2x f(x)= 2 √x-4x

Answer 1

Recall that the x-intercepts of a graph are of the form:

$(x,0)\text{.}$

Since the graph of f(x) passes through (0,0), (1, -2), and (2,0), then its x-intercepts are (0,0) and (2,0).

Therefore f(x) must be as follows:

$\begin{gathered} f(x)=k(x-0)(x-2) \\ =kx(x-2)\text{.} \end{gathered}$

Where k is a constant.

Since f(x) passes through (1,-2), then:

$-2=f(1)\text{.}$

Then:

$-2=k\cdot1(1-2)\text{.}$

Simplifying the above equation we get:

$\begin{gathered} -2=k(-1), \\ -2=-k\text{.} \end{gathered}$

Therefore:

$k=2.$

Therefore f(x) could be as follows:

$\begin{gathered} f(x)=2x(x-2) \\ =2x^2-4x\text{.} \end{gathered}$

Answer: First option.