Identifying the Correct Function

Oct 13, 2025 by miodeya.com 127 views

Understanding the Problem

In this problem, we are tasked with identifying the correct function that represents a function with zeros at $-3, -1, 0$ , and $6$ . To solve this, we need to understand the concept of zeros in a function and how they relate to the factors of the function.

Zeros in a Function

A zero of a function is a value of $x$ that makes the function equal to zero. In other words, if $f(x) = 0$ , then $x$ is a zero of the function. The zeros of a function are also known as the roots or solutions of the function.

Factors of a Function

The factors of a function are the expressions that, when multiplied together, give the original function. In the case of a polynomial function, the factors are the linear expressions that, when multiplied together, give the polynomial.

Identifying the Correct Function

To identify the correct function, we need to examine each of the options and determine which one has zeros at $-3, -1, 0$ , and $6$ . We can do this by factoring each option and checking if it has the desired zeros.

Option A: $y = (x - 6)(x + 1)(x + 3)$

Let's factor the first option:

y = (x - 6)(x + 1)(x + 3)

This option has factors $(x - 6)$ , $(x + 1)$ , and $(x + 3)$ . However, it does not have a factor $(x - 0)$ , which is required to have a zero at $x = 0$ . Therefore, this option is not correct.

Option B: $y = x(x - 3)(x - 1)(x + 6)$

Let's factor the second option:

y = x(x - 3)(x - 1)(x + 6)

This option has factors $(x)$ , $(x - 3)$ , $(x - 1)$ , and $(x + 6)$ . However, it does not have a factor $(x + 3)$ , which is required to have a zero at $x = -3$ . Therefore, this option is not correct.

Option C: $y = x(x - 6)(x + 1)(x + 3)$

Let's factor the third option:

y = x(x - 6)(x + 1)(x + 3)

This option has factors $(x)$ , $(x - 6)$ , $(x + 1)$ , and $(x + 3)$ . However, it does not have a factor $(x - 1)$ , which is required to have a zero at $x = 1$ . Therefore, this option is not correct.

Option D: $y = (x - 0)(x - 6)(x + 1)(x + 3)$

Let's factor the fourth option:

y = (x - 0)(x - 6)(x + 1)(x + 3)

This option has factors $(x - 0)$ , $(x - 6)$ , $(x + 1)$ , and $(x + 3)$ . Since $(x - 0) = x$ , this option can be simplified to:

y = x(x - 6)(x + 1)(x + 3)

This option has factors $(x)$ , $(x - 6)$ , $(x + 1)$ , and $(x + 3)$ . Therefore, this option has zeros at $-3, -1, 0$ , and $6$ .

Conclusion

In conclusion, the correct function that represents a function with zeros at $-3, -1, 0$ , and $6$ is:

y = (x - 0)(x - 6)(x + 1)(x + 3)

This function has factors $(x - 0)$ , $(x - 6)$ , $(x + 1)$ , and $(x + 3)$ , which correspond to the desired zeros.

Key Takeaways

Zeros of a function are values of $x$ that make the function equal to zero.
Factors of a function are expressions that, when multiplied together, give the original function.
To identify the correct function, we need to examine each option and determine which one has the desired zeros.

Final Answer

The final answer is:

y = (x - 0)(x - 6)(x + 1)(x + 3)$$<br/> **Q&A: Selecting the Correct Answer - A Function with Zeros at $-3, -1, 0$, and $6$** ==================================================================== **Frequently Asked Questions** --------------------------- ### Q: What is the concept of zeros in a function? A: A zero of a function is a value of $x$ that makes the function equal to zero. In other words, if $f(x) = 0$, then $x$ is a zero of the function. ### Q: How do I identify the zeros of a function? A: To identify the zeros of a function, you need to set the function equal to zero and solve for $x$. This will give you the values of $x$ that make the function equal to zero. ### Q: What is the relationship between factors and zeros of a function? A: The factors of a function are the expressions that, when multiplied together, give the original function. The zeros of a function are the values of $x$ that make the function equal to zero. In other words, the factors of a function correspond to the zeros of the function. ### Q: How do I determine which function has the desired zeros? A: To determine which function has the desired zeros, you need to examine each option and determine which one has the factors that correspond to the desired zeros. ### Q: What is the correct function that represents a function with zeros at $-3, -1, 0$, and $6$? A: The correct function that represents a function with zeros at $-3, -1, 0$, and $6$ is: $$y = (x - 0)(x - 6)(x + 1)(x + 3)

Q: Why is this function correct?

A: This function is correct because it has factors $(x - 0)$ , $(x - 6)$ , $(x + 1)$ , and $(x + 3)$ , which correspond to the desired zeros.

Q: What are the key takeaways from this problem?

A: The key takeaways from this problem are:

Zeros of a function are values of $x$ that make the function equal to zero.
Factors of a function are expressions that, when multiplied together, give the original function.
To identify the correct function, you need to examine each option and determine which one has the desired zeros.

Q: What is the final answer?

A: The final answer is:

y = (x - 0)(x - 6)(x + 1)(x + 3)

Additional Resources

For more information on zeros and factors of a function, see the following resources: