Let's walk through the steps to determine the correct factored form of a quadratic function given its roots.
### Step 1: Understanding the Roots
The roots of a quadratic function are the values of [tex]\(x\)[/tex] for which the function equals zero. In this problem, we are given the roots [tex]\(5\)[/tex] and [tex]\(-3\)[/tex].
### Step 2: Constructing the Factored Form
The general form of a quadratic function can be written as [tex]\(y = (x - r_1)(x - r_2)\)[/tex], where [tex]\(r_1\)[/tex] and [tex]\(r_2\)[/tex] are the roots.
- For root 5, we use the factor [tex]\((x - 5)\)[/tex].
- For root [tex]\(-3\)[/tex], we use the factor [tex]\((x - (-3))\)[/tex] which simplifies to [tex]\((x + 3)\)[/tex].
### Step 3: Writing the Factored Form
Substituting the given roots into the general form, we get:
[tex]\[ y = (x - 5)(x + 3) \][/tex]
### Step 4: Checking Other Options
Let's verify whether any of the provided options match this form:
1. [tex]\(y = (x - 5)(x + 3)\)[/tex] - This matches our derived form.
2. [tex]\(y = (x + 5)(x - 3)\)[/tex] - This does not match our derived form.
3. [tex]\(y = (x + 5)(x + 3)\)[/tex] - This does not match our derived form.
4. [tex]\(y = (x - 5)(x - 3)\)[/tex] - This does not match our derived form.
### Conclusion
The correct factored form of the quadratic function with roots 5 and -3 is:
[tex]\[ \boxed{(x - 5)(x + 3)} \][/tex]
