Select The Correct Answer.What Are The $x$- And $y$-intercepts Of The Function $g(x)=(x+1)\left(x^2-10x+24\right)$?A. $x$-intercepts: $-1, 4, 6$; $y$-intercept: 24 B.
Understanding the Problem
To find the $x$- and $y$-intercepts of a function, we need to understand what these intercepts represent. The $x$-intercepts are the points where the graph of the function crosses the $x$-axis, meaning the $y$-coordinate is zero. On the other hand, the $y$-intercept is the point where the graph crosses the $y$-axis, meaning the $x$-coordinate is zero.
The Given Function
The given function is $g(x)=(x+1)\left(x^2-10x+24\right)$. To find the $x$- and $y$-intercepts, we need to factorize the function and set it equal to zero.
Factorizing the Function
We can start by factorizing the quadratic expression $x^2-10x+24$. This can be factored as $(x-4)(x-6)$. Therefore, the function can be written as $g(x)=(x+1)(x-4)(x-6)$.
Finding the $x$-Intercepts
To find the $x$-intercepts, we need to set the function equal to zero and solve for $x$. This can be done by setting each factor equal to zero and solving for $x$. Therefore, we have:
Finding the $y$-Intercept
To find the $y$-intercept, we need to substitute $x=0$ into the function and solve for $y$. Therefore, we have:
Conclusion
In conclusion, the $x$-intercepts of the function $g(x)=(x+1)\left(x^2-10x+24\right)$ are $-1, 4, 6$, and the $y$-intercept is $24$. This can be verified by substituting these values into the function and checking if the result is zero for the $x$-intercepts and if the result is a non-zero value for the $y$-intercept.
Answer
The correct answer is A. $x$-intercepts: $-1, 4, 6$; $y$-intercept: 24.
Discussion
This problem requires the application of algebraic techniques to factorize the function and find the intercepts. The factorization of the quadratic expression is a crucial step in finding the $x$-intercepts. Additionally, the substitution of $x=0$ into the function is a straightforward step in finding the $y$-intercept.
Tips and Tricks
- When factorizing a quadratic expression, look for two numbers whose product is the constant term and whose sum is the coefficient of the linear term.
- When finding the $x$-intercepts, set each factor equal to zero and solve for $x$.
- When finding the $y$-intercept, substitute $x=0$ into the function and solve for $y$.
Related Problems
- Find the $x$- and $y$-intercepts of the function $f(x)=(x-2)(x^2+4x+4)$.
- Find the $x$- and $y$-intercepts of the function $h(x)=(x+3)(x^2-2x+1)$.
Further Reading
- For more information on factorizing quadratic expressions, see [1].
- For more information on finding $x$- and $y$-intercepts, see [2].
References
[1] Khan Academy. (n.d.). Factoring Quadratic Expressions. Retrieved from https://www.khanacademy.org/math/algebra/x2factors/x2factors
[2] Math Open Reference. (n.d.). Intercepts. Retrieved from https://www.mathopenref.com/intercepts.html
Understanding the Basics
Before we dive into the questions and answers, let's quickly review the basics of finding $x$- and $y$-intercepts of a function.
- The $x$-intercepts are the points where the graph of the function crosses the $x$-axis, meaning the $y$-coordinate is zero.
- The $y$-intercept is the point where the graph crosses the $y$-axis, meaning the $x$-coordinate is zero.
Q&A
Q1: What is the first step in finding the $x$-intercepts of a function?
A1: The first step in finding the $x$-intercepts of a function is to factorize the function and set it equal to zero.
Q2: How do I factorize a quadratic expression?
A2: To factorize a quadratic expression, look for two numbers whose product is the constant term and whose sum is the coefficient of the linear term.
Q3: What is the next step after factorizing the function?
A3: After factorizing the function, set each factor equal to zero and solve for $x$ to find the $x$-intercepts.
Q4: How do I find the $y$-intercept of a function?
A4: To find the $y$-intercept of a function, substitute $x=0$ into the function and solve for $y$.
Q5: What is the difference between the $x$-intercepts and the $y$-intercept?
A5: The $x$-intercepts are the points where the graph of the function crosses the $x$-axis, while the $y$-intercept is the point where the graph crosses the $y$-axis.
Q6: Can I use the same method to find the $x$- and $y$-intercepts of a rational function?
A6: Yes, you can use the same method to find the $x$- and $y$-intercepts of a rational function. However, you may need to use additional techniques, such as canceling out common factors.
Q7: What is the importance of finding the $x$- and $y$-intercepts of a function?
A7: Finding the $x$- and $y$-intercepts of a function is important because it helps you understand the behavior of the function and its graph.
Q8: Can I use technology, such as a graphing calculator, to find the $x$- and $y$-intercepts of a function?
A8: Yes, you can use technology, such as a graphing calculator, to find the $x$- and $y$-intercepts of a function. However, it's always a good idea to verify your results using algebraic methods.
Additional Tips and Tricks
- When factorizing a quadratic expression, look for two numbers whose product is the constant term and whose sum is the coefficient of the linear term.
- When finding the $x$-intercepts, set each factor equal to zero and solve for $x$.
- When finding the $y$-intercept, substitute $x=0$ into the function and solve for $y$.
- Use technology, such as a graphing calculator, to verify your results and gain a deeper understanding of the function.
Related Problems
- Find the $x$- and $y$-intercepts of the function $f(x)=(x-2)(x^2+4x+4)$.
- Find the $x$- and $y$-intercepts of the function $h(x)=(x+3)(x^2-2x+1)$.
Further Reading
- For more information on factorizing quadratic expressions, see [1].
- For more information on finding $x$- and $y$-intercepts, see [2].
References
[1] Khan Academy. (n.d.). Factoring Quadratic Expressions. Retrieved from https://www.khanacademy.org/math/algebra/x2factors/x2factors
[2] Math Open Reference. (n.d.). Intercepts. Retrieved from https://www.mathopenref.com/intercepts.html