Which Statement Best Describes The Effect On The Graph Of $y=(x-9)^2$ If The Equation Is Changed To $y=(x+9)^2$?A. The Graph Moves Up 18 Units.B. The Graph Moves Down 18 Units.C. The Graph Moves Left 18 Units.D. The Graph Moves

by ADMIN 232 views

Introduction

When dealing with quadratic equations, understanding how changes in the equation affect the graph is crucial. In this article, we will explore the effect of changing the equation $y=(x-9)^2$ to $y=(x+9)^2$ on the graph.

The Original Equation: $y=(x-9)^2$

The original equation is a quadratic equation in the form of $y=(x-h)^2$, where hh is the x-coordinate of the vertex of the parabola. In this case, the vertex is at x=9x=9. The graph of this equation is a parabola that opens upwards, with its vertex at the point (9,0)(9, 0).

The New Equation: $y=(x+9)^2$

The new equation is also a quadratic equation in the form of $y=(x-h)^2$, but with a different value of hh. In this case, the value of hh is −9-9, which means the vertex of the parabola is at x=−9x=-9. The graph of this equation is also a parabola that opens upwards, but its vertex is now at the point (−9,0)(-9, 0).

Comparing the Two Equations

To understand the effect of changing the equation from $y=(x-9)^2$ to $y=(x+9)^2$, we need to compare the two graphs. The key difference between the two equations is the value of hh. In the original equation, h=9h=9, while in the new equation, h=−9h=-9.

The Effect on the Graph

When we change the equation from $y=(x-9)^2$ to $y=(x+9)^2$, the graph of the parabola shifts horizontally. The vertex of the parabola moves from x=9x=9 to x=−9x=-9, which means the graph moves left by 18 units.

Conclusion

In conclusion, when the equation $y=(x-9)^2$ is changed to $y=(x+9)^2$, the graph of the parabola moves left by 18 units. This is because the value of hh changes from 9 to -9, causing the vertex of the parabola to move from x=9x=9 to x=−9x=-9.

Answer

The correct answer is:

  • C. The graph moves left 18 units.

Why is this the correct answer?

This is the correct answer because the change in the equation from $y=(x-9)^2$ to $y=(x+9)^2$ causes the graph of the parabola to shift horizontally. The vertex of the parabola moves from x=9x=9 to x=−9x=-9, which means the graph moves left by 18 units.

What is the significance of this result?

This result is significant because it shows how changes in the equation of a quadratic function can affect the graph of the function. Understanding how changes in the equation affect the graph is crucial in mathematics and has many practical applications in fields such as physics, engineering, and economics.

What are some real-world applications of this result?

Some real-world applications of this result include:

  • Physics: In physics, the equation of a quadratic function can be used to model the motion of an object under the influence of a force. Understanding how changes in the equation affect the graph can help physicists predict the motion of an object.
  • Engineering: In engineering, the equation of a quadratic function can be used to design curves and surfaces. Understanding how changes in the equation affect the graph can help engineers design more efficient and effective curves and surfaces.
  • Economics: In economics, the equation of a quadratic function can be used to model the behavior of economic systems. Understanding how changes in the equation affect the graph can help economists predict the behavior of economic systems.

What are some common mistakes to avoid when working with quadratic equations?

Some common mistakes to avoid when working with quadratic equations include:

  • Not checking the domain of the function: When working with quadratic equations, it is essential to check the domain of the function to ensure that it is defined for all values of x.
  • Not checking the range of the function: When working with quadratic equations, it is essential to check the range of the function to ensure that it is defined for all values of y.
  • Not using the correct formula for the vertex: When working with quadratic equations, it is essential to use the correct formula for the vertex to ensure that the vertex is calculated correctly.

What are some tips for working with quadratic equations?

Some tips for working with quadratic equations include:

  • Use the correct formula for the vertex: When working with quadratic equations, it is essential to use the correct formula for the vertex to ensure that the vertex is calculated correctly.
  • Check the domain and range of the function: When working with quadratic equations, it is essential to check the domain and range of the function to ensure that it is defined for all values of x and y.
  • Use a graphing calculator or software: When working with quadratic equations, it is often helpful to use a graphing calculator or software to visualize the graph of the function and check the domain and range of the function.

What are some common applications of quadratic equations?

Some common applications of quadratic equations include:

  • Physics: Quadratic equations are used to model the motion of objects under the influence of a force.
  • Engineering: Quadratic equations are used to design curves and surfaces.
  • Economics: Quadratic equations are used to model the behavior of economic systems.
  • Computer Science: Quadratic equations are used to solve problems in computer science, such as finding the shortest path between two points in a graph.

What are some common mistakes to avoid when working with quadratic equations in real-world applications?

Some common mistakes to avoid when working with quadratic equations in real-world applications include:

  • Not checking the domain and range of the function: When working with quadratic equations in real-world applications, it is essential to check the domain and range of the function to ensure that it is defined for all values of x and y.
  • Not using the correct formula for the vertex: When working with quadratic equations in real-world applications, it is essential to use the correct formula for the vertex to ensure that the vertex is calculated correctly.
  • Not considering the constraints of the problem: When working with quadratic equations in real-world applications, it is essential to consider the constraints of the problem to ensure that the solution is feasible and practical.

What are some tips for working with quadratic equations in real-world applications?

Some tips for working with quadratic equations in real-world applications include:

  • Use a graphing calculator or software: When working with quadratic equations in real-world applications, it is often helpful to use a graphing calculator or software to visualize the graph of the function and check the domain and range of the function.
  • Check the domain and range of the function: When working with quadratic equations in real-world applications, it is essential to check the domain and range of the function to ensure that it is defined for all values of x and y.
  • Use the correct formula for the vertex: When working with quadratic equations in real-world applications, it is essential to use the correct formula for the vertex to ensure that the vertex is calculated correctly.

Conclusion

In conclusion, quadratic equations are a fundamental concept in mathematics and have many practical applications in fields such as physics, engineering, and economics. Understanding how changes in the equation affect the graph is crucial in mathematics and has many real-world applications. By following the tips and avoiding common mistakes outlined in this article, you can work with quadratic equations effectively and solve problems in a variety of fields.