Type The Correct Answer In Each Box. Use Numerals Instead Of Words.The Zeros Of The Function F ( X ) = − ( X + 1 ) ( X − 3 ) ( X + 2 F(x)=-(x+1)(x-3)(x+2 F ( X ) = − ( X + 1 ) ( X − 3 ) ( X + 2 ] Are − 1 -1 − 1 , 3 3 3 , And □ \square □ , And The Y Y Y -intercept Of The Function Is Located At $(0,

Mar 9, 2025 by ADMIN 299 views

**Understanding the Zeros and Y-Intercept of a Function**

Introduction

In mathematics, a function is a relation between a set of inputs, called the domain, and a set of possible outputs, called the range. The zeros of a function are the values of x that make the function equal to zero. The y-intercept of a function is the point where the function intersects the y-axis. In this article, we will explore the zeros and y-intercept of a given function and use numerals instead of words to provide the correct answers.

The Function

The given function is $f(x)=-(x+1)(x-3)(x+2)$ . To find the zeros of the function, we need to set the function equal to zero and solve for x.

Finding the Zeros

To find the zeros of the function, we can start by factoring the function:

f(x)=-(x+1)(x-3)(x+2)

We can see that the function is already factored, and we can set each factor equal to zero to find the zeros:

(x+1)=0 \Rightarrow x=-1

(x-3)=0 \Rightarrow x=3

(x+2)=0 \Rightarrow x=-2

Therefore, the zeros of the function are $-1$ , $3$ , and $-2$ .

The Y-Intercept

To find the y-intercept of the function, we need to find the point where the function intersects the y-axis. This occurs when x=0. We can substitute x=0 into the function to find the y-intercept:

f(0)=-(0+1)(0-3)(0+2)

f(0)=-(1)(-3)(2)

f(0)=6

Therefore, the y-intercept of the function is located at $(0,6)$ .

Conclusion

In conclusion, the zeros of the function $f(x)=-(x+1)(x-3)(x+2)$ are $-1$ , $3$ , and $-2$ , and the y-intercept of the function is located at $(0,6)$ . We used numerals instead of words to provide the correct answers.

Key Takeaways

The zeros of a function are the values of x that make the function equal to zero.
The y-intercept of a function is the point where the function intersects the y-axis.
To find the zeros of a function, we can set each factor of the function equal to zero and solve for x.
To find the y-intercept of a function, we can substitute x=0 into the function.

Practice Problems

Find the zeros of the function $f(x)=(x+2)(x-1)(x+3)$ .
Find the y-intercept of the function $f(x)=(x-2)(x+1)(x-4)$ .

Answers

The zeros of the function are $-2$ , $1$ , and $-3$ .
The y-intercept of the function is located at $(0,-24)$ .

References

[1] "Algebra" by Michael Artin
[2] "Calculus" by Michael Spivak

Additional Resources

Khan Academy: Algebra
MIT OpenCourseWare: Calculus

Introduction

In our previous article, we explored the zeros and y-intercept of a given function. In this article, we will answer some frequently asked questions about the zeros and y-intercept of a function.

Q: What are the zeros of a function?

A: The zeros of a function are the values of x that make the function equal to zero. In other words, they are the points where the graph of the function intersects the x-axis.

Q: How do I find the zeros of a function?

A: To find the zeros of a function, you can set each factor of the function equal to zero and solve for x. For example, if the function is $f(x)=(x+1)(x-3)(x+2)$ , you can set each factor equal to zero and solve for x:

(x+1)=0 \Rightarrow x=-1

(x-3)=0 \Rightarrow x=3

(x+2)=0 \Rightarrow x=-2

Q: What is the y-intercept of a function?

A: The y-intercept of a function is the point where the function intersects the y-axis. This occurs when x=0. To find the y-intercept of a function, you can substitute x=0 into the function.

Q: How do I find the y-intercept of a function?

A: To find the y-intercept of a function, you can substitute x=0 into the function. For example, if the function is $f(x)=(x+1)(x-3)(x+2)$ , you can substitute x=0 into the function:

f(0)=-(0+1)(0-3)(0+2)

f(0)=-(1)(-3)(2)

f(0)=6

Therefore, the y-intercept of the function is located at $(0,6)$ .

Q: What is the difference between the zeros and y-intercept of a function?

A: The zeros of a function are the points where the graph of the function intersects the x-axis, while the y-intercept of a function is the point where the graph of the function intersects the y-axis.

Q: Can you provide an example of a function with zeros and a y-intercept?

A: Yes, consider the function $f(x)=(x+1)(x-3)(x+2)$ . The zeros of the function are $-1$ , $3$ , and $-2$ , and the y-intercept of the function is located at $(0,6)$ .

Q: How do I use the zeros and y-intercept of a function in real-world applications?

A: The zeros and y-intercept of a function can be used in a variety of real-world applications, such as:

Modeling population growth and decline
Analyzing the behavior of electrical circuits
Predicting the behavior of physical systems, such as the motion of objects under the influence of gravity

Q: What are some common mistakes to avoid when finding the zeros and y-intercept of a function?

A: Some common mistakes to avoid when finding the zeros and y-intercept of a function include:

Failing to factor the function correctly
Failing to set each factor equal to zero and solve for x
Failing to substitute x=0 into the function to find the y-intercept