Find The \[$ X \$\]-intercept(s) And The Coordinates Of The Vertex For The Parabola \[$ Y = -x^2 + 4x - 4 \$\].If There Is More Than One \[$ X \$\]-intercept, Separate Them With Commas.\[$ X \$\]-intercepts:\[$

Feb 28, 2025 by ADMIN 211 views

Introduction

In mathematics, a parabola is a type of quadratic equation that can be represented in the form of y = ax^2 + bx + c, where a, b, and c are constants. The x-intercept(s) of a parabola are the points where the parabola intersects the x-axis, and the vertex is the lowest or highest point on the parabola. In this article, we will focus on finding the x-intercept(s) and the coordinates of the vertex for the parabola y = -x^2 + 4x - 4.

Finding the x-Intercept(s)

To find the x-intercept(s) of a parabola, we need to set y = 0 and solve for x. This is because the x-intercept(s) occur when the parabola intersects the x-axis, and at these points, the value of y is equal to 0.

For the parabola y = -x^2 + 4x - 4, we can set y = 0 and solve for x as follows:

0 = -x^2 + 4x - 4

To solve this equation, we can use the quadratic formula:

x = (-b ± √(b^2 - 4ac)) / 2a

In this case, a = -1, b = 4, and c = -4. Plugging these values into the quadratic formula, we get:

x = (-(4) ± √((4)^2 - 4(-1)(-4))) / 2(-1) x = (-4 ± √(16 - 16)) / -2 x = (-4 ± √0) / -2 x = (-4 ± 0) / -2 x = -4 / -2 x = 2

Therefore, the x-intercept(s) of the parabola y = -x^2 + 4x - 4 is 2.

Finding the Coordinates of the Vertex

The vertex of a parabola is the lowest or highest point on the parabola. To find the coordinates of the vertex, we need to find the x-coordinate of the vertex and then plug this value into the equation of the parabola to find the corresponding y-coordinate.

The x-coordinate of the vertex can be found using the formula:

x = -b / 2a

In this case, a = -1 and b = 4. Plugging these values into the formula, we get:

x = -4 / 2(-1) x = -4 / -2 x = 2

Now that we have the x-coordinate of the vertex, we can plug this value into the equation of the parabola to find the corresponding y-coordinate:

y = -x^2 + 4x - 4 y = -(2)^2 + 4(2) - 4 y = -4 + 8 - 4 y = 0

Therefore, the coordinates of the vertex of the parabola y = -x^2 + 4x - 4 is (2, 0).

Conclusion

In this article, we have found the x-intercept(s) and the coordinates of the vertex for the parabola y = -x^2 + 4x - 4. The x-intercept(s) is 2, and the coordinates of the vertex is (2, 0). These values can be used to graph the parabola and to understand its behavior.

Graphing the Parabola

To graph the parabola y = -x^2 + 4x - 4, we can use the x-intercept(s) and the coordinates of the vertex. The x-intercept(s) is 2, and the coordinates of the vertex is (2, 0). We can plot these points on a coordinate plane and then draw a smooth curve through them to represent the parabola.

Real-World Applications

The parabola y = -x^2 + 4x - 4 has many real-world applications. For example, it can be used to model the trajectory of a projectile, such as a thrown ball or a rocket. It can also be used to model the shape of a satellite dish or a parabolic mirror.

Final Thoughts

In conclusion, finding the x-intercept(s) and the coordinates of the vertex for a parabola is an important concept in mathematics. It can be used to graph the parabola and to understand its behavior. The parabola y = -x^2 + 4x - 4 has many real-world applications, and it can be used to model the trajectory of a projectile or the shape of a satellite dish or a parabolic mirror.

References

[1] "Quadratic Equations" by Math Open Reference
[2] "Parabolas" by Math Is Fun
[3] "Graphing Quadratic Equations" by Purplemath

Discussion

What are some real-world applications of the parabola y = -x^2 + 4x - 4?
How can the x-intercept(s) and the coordinates of the vertex be used to graph the parabola?
What are some other ways to find the x-intercept(s) and the coordinates of the vertex for a parabola?

Introduction

In our previous article, we discussed how to find the x-intercept(s) and the coordinates of the vertex for the parabola y = -x^2 + 4x - 4. In this article, we will answer some frequently asked questions related to this topic.

Q&A

Q: What is the x-intercept(s) of the parabola y = -x^2 + 4x - 4?

A: The x-intercept(s) of the parabola y = -x^2 + 4x - 4 is 2.

Q: How do I find the x-intercept(s) of a parabola?

A: To find the x-intercept(s) of a parabola, you need to set y = 0 and solve for x. This is because the x-intercept(s) occur when the parabola intersects the x-axis, and at these points, the value of y is equal to 0.

Q: What is the vertex of the parabola y = -x^2 + 4x - 4?

A: The vertex of the parabola y = -x^2 + 4x - 4 is (2, 0).

Q: How do I find the coordinates of the vertex of a parabola?

A: To find the coordinates of the vertex of a parabola, you need to find the x-coordinate of the vertex and then plug this value into the equation of the parabola to find the corresponding y-coordinate.

Q: What are some real-world applications of the parabola y = -x^2 + 4x - 4?

A: The parabola y = -x^2 + 4x - 4 has many real-world applications, such as modeling the trajectory of a projectile, the shape of a satellite dish or a parabolic mirror.

Q: How can I graph the parabola y = -x^2 + 4x - 4?

A: To graph the parabola y = -x^2 + 4x - 4, you can use the x-intercept(s) and the coordinates of the vertex. Plot these points on a coordinate plane and then draw a smooth curve through them to represent the parabola.

Q: What are some other ways to find the x-intercept(s) and the coordinates of the vertex for a parabola?

A: There are several other ways to find the x-intercept(s) and the coordinates of the vertex for a parabola, such as using the quadratic formula or factoring the equation.

Conclusion

In this article, we have answered some frequently asked questions related to finding the x-intercept(s) and the coordinates of the vertex for the parabola y = -x^2 + 4x - 4. We hope that this article has been helpful in understanding this concept.

Final Thoughts

Finding the x-intercept(s) and the coordinates of the vertex for a parabola is an important concept in mathematics. It can be used to graph the parabola and to understand its behavior. The parabola y = -x^2 + 4x - 4 has many real-world applications, and it can be used to model the trajectory of a projectile or the shape of a satellite dish or a parabolic mirror.

References

[1] "Quadratic Equations" by Math Open Reference
[2] "Parabolas" by Math Is Fun
[3] "Graphing Quadratic Equations" by Purplemath

Discussion

What are some other ways to find the x-intercept(s) and the coordinates of the vertex for a parabola?
How can the x-intercept(s) and the coordinates of the vertex be used to graph the parabola?
What are some real-world applications of the parabola y = -x^2 + 4x - 4?

Find The \[$ X \$\]-intercept(s) And The Coordinates Of The Vertex For The Parabola \[$ Y = -x^2 + 4x - 4 \$\].If There Is More Than One \[$ X \$\]-intercept, Separate Them With Commas.\[$ X \$\]-intercepts:\[$

Introduction

Finding the x-Intercept(s)

Finding the Coordinates of the Vertex

Conclusion

Graphing the Parabola

Real-World Applications

Final Thoughts

References

Discussion

Related Topics

Introduction

Q&A

Q: What is the x-intercept(s) of the parabola y = -x^2 + 4x - 4?

Q: How do I find the x-intercept(s) of a parabola?

Q: What is the vertex of the parabola y = -x^2 + 4x - 4?

Q: How do I find the coordinates of the vertex of a parabola?

Q: What are some real-world applications of the parabola y = -x^2 + 4x - 4?

Q: How can I graph the parabola y = -x^2 + 4x - 4?

Q: What are some other ways to find the x-intercept(s) and the coordinates of the vertex for a parabola?

Conclusion

Final Thoughts

References

Discussion

Related Topics