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.

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Understanding the Problem

To find the $x$- and $y$-intercepts of a function, we need to understand what these intercepts represent. The $x$-intercept is the point where the graph of the function crosses the $x$-axis, and the $y$-intercept is the point where the graph crosses the $y$-axis. In other words, the $x$-intercept is the value of $x$ where $y=0$, and the $y$-intercept is the value of $y$ where $x=0$.

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 set the function equal to zero and solve for $x$.

Finding the $x$-Intercepts

To find the $x$-intercepts, we set the function equal to zero and solve for $x$. We can start by factoring the quadratic expression inside the parentheses:

g(x)=(x+1)(x2−10x+24)g(x)=(x+1)\left(x^2-10x+24\right)

g(x)=(x+1)(x−4)(x−6)g(x)=(x+1)(x-4)(x-6)

Now, we can set the function equal to zero and solve for $x$:

(x+1)(x−4)(x−6)=0(x+1)(x-4)(x-6)=0

This equation has three solutions: $x=-1$, $x=4$, and $x=6$. These are the $x$-intercepts of the function.

Finding the $y$-Intercept

To find the $y$-intercept, we need to find the value of $y$ when $x=0$. We can substitute $x=0$ into the function:

g(0)=(0+1)(02−10(0)+24)g(0)=(0+1)\left(0^2-10(0)+24\right)

g(0)=(1)(24)g(0)=(1)(24)

g(0)=24g(0)=24

So, the $y$-intercept of the function is $24$.

Conclusion

In conclusion, the $x$-intercepts of the function $g(x)=(x+1)\left(x^2-10x+24\right)$ are $-1$, $4$, and $6$, and the $y$-intercept is $24$. These intercepts can be found by setting the function equal to zero and solving for $x$, and by substituting $x=0$ into the function.

Answer

The correct answer is A. $x$-intercepts: $-1, 4, 6$; $y$-intercept: 24.

Discussion

This problem requires the use of algebraic techniques, such as factoring and solving quadratic equations. It also requires an understanding of the concept of intercepts and how to find them. The solution to this problem can be applied to a variety of real-world situations, such as finding the points of intersection between two curves or the maximum or minimum values of a function.

Related Topics

  • Finding the $x$- and $y$-intercepts of a linear function
  • Finding the $x$- and $y$-intercepts of a quadratic function
  • Graphing functions and finding their intercepts
  • Solving quadratic equations and factoring

Practice Problems

  • Find the $x$- and $y$-intercepts of the function $f(x)=(x-2)\left(x^2+5x+6\right)$.
  • Find the $x$- and $y$-intercepts of the function $h(x)=(x+3)\left(x^2-2x-8\right)$.
  • Find the $x$- and $y$-intercepts of the function $j(x)=(x-1)\left(x^2+4x+4\right)$.

Conclusion

In conclusion, finding the $x$- and $y$-intercepts of a function is an important concept in algebra and mathematics. It requires the use of algebraic techniques, such as factoring and solving quadratic equations, and an understanding of the concept of intercepts. The solution to this problem can be applied to a variety of real-world situations, such as finding the points of intersection between two curves or the maximum or minimum values of a function.

Q: What are the $x$- and $y$-intercepts of a function?

A: The $x$-intercept is the point where the graph of the function crosses the $x$-axis, and the $y$-intercept is the point where the graph crosses the $y$-axis. In other words, the $x$-intercept is the value of $x$ where $y=0$, and the $y$-intercept is the value of $y$ where $x=0$.

Q: How do I find the $x$-intercepts of a function?

A: To find the $x$-intercepts, you need to set the function equal to zero and solve for $x$. You can start by factoring the function, if possible, and then set each factor equal to zero and solve for $x$.

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

A: To find the $y$-intercept, you need to find the value of $y$ when $x=0$. You can substitute $x=0$ into the function and solve for $y$.

Q: What is the difference between the $x$- and $y$-intercepts?

A: The $x$-intercept is the point where the graph of the function crosses the $x$-axis, and the $y$-intercept is the point where the graph crosses the $y$-axis. In other words, the $x$-intercept is the value of $x$ where $y=0$, and the $y$-intercept is the value of $y$ where $x=0$.

Q: Can I find the $x$- and $y$-intercepts of a function using a graphing calculator?

A: Yes, you can find the $x$- and $y$-intercepts of a function using a graphing calculator. You can graph the function and use the calculator to find the points of intersection with the $x$- and $y$-axes.

Q: What are some common mistakes to avoid when finding the $x$- and $y$-intercepts of a function?

A: Some common mistakes to avoid when finding the $x$- and $y$-intercepts of a function include:

  • Not setting the function equal to zero when finding the $x$-intercepts
  • Not substituting $x=0$ into the function when finding the $y$-intercept
  • Not factoring the function, if possible, when finding the $x$-intercepts
  • Not using a graphing calculator to check the points of intersection with the $x$- and $y$-axes

Q: How do I apply the concept of $x$- and $y$-intercepts to real-world problems?

A: The concept of $x$- and $y$-intercepts can be applied to a variety of real-world problems, such as:

  • Finding the points of intersection between two curves or surfaces
  • Determining the maximum or minimum values of a function
  • Analyzing the behavior of a function over a given interval
  • Solving optimization problems

Q: What are some advanced topics related to finding the $x$- and $y$-intercepts of a function?

A: Some advanced topics related to finding the $x$- and $y$-intercepts of a function include:

  • Finding the $x$- and $y$-intercepts of a function with complex coefficients
  • Finding the $x$- and $y$-intercepts of a function with multiple variables
  • Finding the $x$- and $y$-intercepts of a function with a non-linear relationship between the variables
  • Using numerical methods to find the $x$- and $y$-intercepts of a function

Q: How do I practice finding the $x$- and $y$-intercepts of a function?

A: You can practice finding the $x$- and $y$-intercepts of a function by:

  • Working through example problems in a textbook or online resource
  • Using a graphing calculator to visualize the function and find the points of intersection with the $x$- and $y$-axes
  • Creating your own example problems and solving them
  • Joining a study group or online community to discuss and practice finding the $x$- and $y$-intercepts of a function.