Which Of The Following Is The Graph Of $y = -(x - 2)^3 - 5$?

by ADMIN 63 views

Introduction

Graphing a cubic function can be a complex task, but with the right approach, it can be broken down into manageable steps. In this article, we will explore the graph of the cubic function y=−(x−2)3−5y = -(x - 2)^3 - 5. We will analyze the equation, identify its key features, and determine the graph that represents this function.

Understanding the Equation

The given equation is y=−(x−2)3−5y = -(x - 2)^3 - 5. To understand this equation, let's break it down into its components. The equation is a cubic function, which means it has a degree of 3. The general form of a cubic function is y=ax3+bx2+cx+dy = ax^3 + bx^2 + cx + d, where aa, bb, cc, and dd are constants.

In this case, the equation can be rewritten as y=−(x−2)3−5y = -(x - 2)^3 - 5. The negative sign in front of the expression (x−2)3(x - 2)^3 indicates that the graph will be reflected across the x-axis. The expression (x−2)3(x - 2)^3 represents a translation of the graph 2 units to the right. Finally, the constant term −5-5 represents a vertical translation of the graph 5 units down.

Identifying Key Features

To identify the key features of the graph, we need to analyze the equation. The equation has a negative leading coefficient, which means the graph will be reflected across the x-axis. The expression (x−2)3(x - 2)^3 represents a translation of the graph 2 units to the right. The constant term −5-5 represents a vertical translation of the graph 5 units down.

Graphing the Function

To graph the function, we can start by plotting the vertex of the graph. The vertex is the point on the graph where the function changes from decreasing to increasing or vice versa. In this case, the vertex is at the point (2,−5)(2, -5).

Next, we can plot the x-intercepts of the graph. The x-intercepts are the points on the graph where the function crosses the x-axis. In this case, the x-intercepts are at the points (−1,0)(-1, 0) and (3,0)(3, 0).

Finally, we can plot the y-intercept of the graph. The y-intercept is the point on the graph where the function crosses the y-axis. In this case, the y-intercept is at the point (0,−125)(0, -125).

Determining the Graph

Based on the key features and the graph we have plotted, we can determine the graph that represents the function y=−(x−2)3−5y = -(x - 2)^3 - 5. The graph is a cubic function with a negative leading coefficient, which means it is reflected across the x-axis. The graph has a translation of 2 units to the right and a vertical translation of 5 units down.

Conclusion

In conclusion, the graph of the cubic function y=−(x−2)3−5y = -(x - 2)^3 - 5 is a reflection of the graph of the function y=(x−2)3y = (x - 2)^3 across the x-axis, with a translation of 2 units to the right and a vertical translation of 5 units down. The graph has a vertex at the point (2,−5)(2, -5), x-intercepts at the points (−1,0)(-1, 0) and (3,0)(3, 0), and a y-intercept at the point (0,−125)(0, -125).

Graph Options

There are several graph options that could represent the function y=−(x−2)3−5y = -(x - 2)^3 - 5. However, based on the key features and the graph we have plotted, the most likely graph is:

  • Graph A: A cubic function with a negative leading coefficient, reflected across the x-axis, with a translation of 2 units to the right and a vertical translation of 5 units down.

Graph A

Graph A is a cubic function with a negative leading coefficient, reflected across the x-axis, with a translation of 2 units to the right and a vertical translation of 5 units down. The graph has a vertex at the point (2,−5)(2, -5), x-intercepts at the points (−1,0)(-1, 0) and (3,0)(3, 0), and a y-intercept at the point (0,−125)(0, -125).

Graph B

Graph B is a cubic function with a positive leading coefficient, not reflected across the x-axis, with a translation of 2 units to the right and a vertical translation of 5 units down. The graph has a vertex at the point (2,−5)(2, -5), x-intercepts at the points (−1,0)(-1, 0) and (3,0)(3, 0), and a y-intercept at the point (0,−125)(0, -125).

Graph C

Graph C is a cubic function with a negative leading coefficient, reflected across the x-axis, with a translation of 2 units to the left and a vertical translation of 5 units down. The graph has a vertex at the point (2,−5)(2, -5), x-intercepts at the points (−1,0)(-1, 0) and (3,0)(3, 0), and a y-intercept at the point (0,−125)(0, -125).

Graph D

Graph D is a cubic function with a positive leading coefficient, not reflected across the x-axis, with a translation of 2 units to the left and a vertical translation of 5 units down. The graph has a vertex at the point (2,−5)(2, -5), x-intercepts at the points (−1,0)(-1, 0) and (3,0)(3, 0), and a y-intercept at the point (0,−125)(0, -125).

Conclusion

In conclusion, the graph of the cubic function y=−(x−2)3−5y = -(x - 2)^3 - 5 is a reflection of the graph of the function y=(x−2)3y = (x - 2)^3 across the x-axis, with a translation of 2 units to the right and a vertical translation of 5 units down. The graph has a vertex at the point (2,−5)(2, -5), x-intercepts at the points (−1,0)(-1, 0) and (3,0)(3, 0), and a y-intercept at the point (0,−125)(0, -125).

Graph Options Comparison

Graph Description Vertex X-Intercepts Y-Intercept
A Cubic function with a negative leading coefficient, reflected across the x-axis, with a translation of 2 units to the right and a vertical translation of 5 units down. (2, -5) (-1, 0), (3, 0) (0, -125)
B Cubic function with a positive leading coefficient, not reflected across the x-axis, with a translation of 2 units to the right and a vertical translation of 5 units down. (2, -5) (-1, 0), (3, 0) (0, -125)
C Cubic function with a negative leading coefficient, reflected across the x-axis, with a translation of 2 units to the left and a vertical translation of 5 units down. (2, -5) (-1, 0), (3, 0) (0, -125)
D Cubic function with a positive leading coefficient, not reflected across the x-axis, with a translation of 2 units to the left and a vertical translation of 5 units down. (2, -5) (-1, 0), (3, 0) (0, -125)

Conclusion

In conclusion, the graph of the cubic function y=−(x−2)3−5y = -(x - 2)^3 - 5 is a reflection of the graph of the function y=(x−2)3y = (x - 2)^3 across the x-axis, with a translation of 2 units to the right and a vertical translation of 5 units down. The graph has a vertex at the point (2,−5)(2, -5), x-intercepts at the points (−1,0)(-1, 0) and (3,0)(3, 0), and a y-intercept at the point (0,−125)(0, -125).

Graph A is the Correct Answer

Introduction

In our previous article, we explored the graph of the cubic function y=−(x−2)3−5y = -(x - 2)^3 - 5. We analyzed the equation, identified its key features, and determined the graph that represents this function. In this article, we will answer some frequently asked questions about the graph of y=−(x−2)3−5y = -(x - 2)^3 - 5.

Q: What is the vertex of the graph of y=−(x−2)3−5y = -(x - 2)^3 - 5?

A: The vertex of the graph of y=−(x−2)3−5y = -(x - 2)^3 - 5 is at the point (2,−5)(2, -5).

Q: What are the x-intercepts of the graph of y=−(x−2)3−5y = -(x - 2)^3 - 5?

A: The x-intercepts of the graph of y=−(x−2)3−5y = -(x - 2)^3 - 5 are at the points (−1,0)(-1, 0) and (3,0)(3, 0).

Q: What is the y-intercept of the graph of y=−(x−2)3−5y = -(x - 2)^3 - 5?

A: The y-intercept of the graph of y=−(x−2)3−5y = -(x - 2)^3 - 5 is at the point (0,−125)(0, -125).

Q: Is the graph of y=−(x−2)3−5y = -(x - 2)^3 - 5 a reflection of the graph of y=(x−2)3y = (x - 2)^3 across the x-axis?

A: Yes, the graph of y=−(x−2)3−5y = -(x - 2)^3 - 5 is a reflection of the graph of y=(x−2)3y = (x - 2)^3 across the x-axis.

Q: What is the translation of the graph of y=−(x−2)3−5y = -(x - 2)^3 - 5?

A: The graph of y=−(x−2)3−5y = -(x - 2)^3 - 5 has a translation of 2 units to the right.

Q: What is the vertical translation of the graph of y=−(x−2)3−5y = -(x - 2)^3 - 5?

A: The graph of y=−(x−2)3−5y = -(x - 2)^3 - 5 has a vertical translation of 5 units down.

Q: What is the degree of the graph of y=−(x−2)3−5y = -(x - 2)^3 - 5?

A: The degree of the graph of y=−(x−2)3−5y = -(x - 2)^3 - 5 is 3.

Q: Is the graph of y=−(x−2)3−5y = -(x - 2)^3 - 5 a cubic function?

A: Yes, the graph of y=−(x−2)3−5y = -(x - 2)^3 - 5 is a cubic function.

Conclusion

In conclusion, the graph of y=−(x−2)3−5y = -(x - 2)^3 - 5 is a reflection of the graph of y=(x−2)3y = (x - 2)^3 across the x-axis, with a translation of 2 units to the right and a vertical translation of 5 units down. The graph has a vertex at the point (2,−5)(2, -5), x-intercepts at the points (−1,0)(-1, 0) and (3,0)(3, 0), and a y-intercept at the point (0,−125)(0, -125).

Frequently Asked Questions

  • What is the vertex of the graph of y=−(x−2)3−5y = -(x - 2)^3 - 5?
  • What are the x-intercepts of the graph of y=−(x−2)3−5y = -(x - 2)^3 - 5?
  • What is the y-intercept of the graph of y=−(x−2)3−5y = -(x - 2)^3 - 5?
  • Is the graph of y=−(x−2)3−5y = -(x - 2)^3 - 5 a reflection of the graph of y=(x−2)3y = (x - 2)^3 across the x-axis?
  • What is the translation of the graph of y=−(x−2)3−5y = -(x - 2)^3 - 5?
  • What is the vertical translation of the graph of y=−(x−2)3−5y = -(x - 2)^3 - 5?
  • What is the degree of the graph of y=−(x−2)3−5y = -(x - 2)^3 - 5?
  • Is the graph of y=−(x−2)3−5y = -(x - 2)^3 - 5 a cubic function?

Answers

  • The vertex of the graph of y=−(x−2)3−5y = -(x - 2)^3 - 5 is at the point (2,−5)(2, -5).
  • The x-intercepts of the graph of y=−(x−2)3−5y = -(x - 2)^3 - 5 are at the points (−1,0)(-1, 0) and (3,0)(3, 0).
  • The y-intercept of the graph of y=−(x−2)3−5y = -(x - 2)^3 - 5 is at the point (0,−125)(0, -125).
  • Yes, the graph of y=−(x−2)3−5y = -(x - 2)^3 - 5 is a reflection of the graph of y=(x−2)3y = (x - 2)^3 across the x-axis.
  • The graph of y=−(x−2)3−5y = -(x - 2)^3 - 5 has a translation of 2 units to the right.
  • The graph of y=−(x−2)3−5y = -(x - 2)^3 - 5 has a vertical translation of 5 units down.
  • The degree of the graph of y=−(x−2)3−5y = -(x - 2)^3 - 5 is 3.
  • Yes, the graph of y=−(x−2)3−5y = -(x - 2)^3 - 5 is a cubic function.