To answer the question in the image, we need to determine the correct function in factored form with zeros at the origin (0, 0) and at (4, 0).

Step-by-Step Analysis:

1. Zeros of the Function:
   A function has zeros at a givenThe problem point is asking for a function in factored form with zeros at the origin $(0, 0)$ and at $(4, 0)$. if Let's analyze the given choices substit:

Key Information:

uting- A zero at $x = 0 those$ implies values the factor (x\ for).

• A zero at \(x = 4 implies the factorx $(x -$ makes4 the function equal to)\ zero.). The- The correct function will multiply these factors problem states together the.

zeros Analysis are of at:

Options : -1. \(x =f(x 0 (the origin) -) = (x = x ^4\2 +)

2. Factored Form x +:
   4 The\ factored): form of a polynomial with zeros at This is not factored form $,x = and0$ and $x it = does not4$ produce can zeros be at written as (: x = \ 0) and $x = 4$.

3. (f(x)[ f(x) = x(x - 4) ]

4. Matching the Answer:
   Among the provided options:

   • $f(x) = x(x - 4)$ matches the correct factored form based on the given zeros.

Correct Answer:

The correct choice is: = x(x\ - 4)): This is in factored form[ and has zerosf at $x(x = )0$ and $x = 4 =$. 3. (f(x) x = (x + 4)^2(x): This has a double root at - (x = 4 -) \4]

), notWould at you like me to elaborate on the reasoning or show ( graphing steps to confirm? Let me know!

xHere = are 05 related questions):

1. What or is the x expanded form = of \(f(x) = x(x - 4)? 24.). How4. would $f additional zeros(x affect) the = x fact(x + ored4)$: form This has? zeros at (x3. Can a = quadratic0 function\ have) only one and (x = -4 zero? ), which does not4. What is the vertex of the parab match the requiredola zeros.

defined### by ( Correctf(x Answer) =: x(xThe - correct function is: 4$f(x) = x(x - 4)$

Let me know if you want further details or have))? questions! Here are some related questions for practice:

1. What5 happens if a factor like ((x. - 4)^2\ How) is included in a function?
2. How can we identify the multiplicity of a root from factored form can you determine the axis?
3. What is of the graph of $f(x) symmetry = x(x - 4)$?
4. How would you rewrite (f(x)\ for) in expanded form? 5 a. What quadratic would the function look like if there was an additional root at $x = -3$?

Tip: function Always match the?

zeros of the function to their respective factors!Tip: Always cross-check your zeros and their placement in the factored form to avoid sign errors!