Find The \[$x\$\]-intercept(s) And The Coordinates Of The Vertex For The Parabola \[$y = X^2 + 2x - 8\$\]. If There Is More Than One \[$x\$\]-intercept, Separate Them With Commas.

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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 + 2x - 8.

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 + 2x - 8, we can set y = 0 and solve for x as follows:

0 = x^2 + 2x - 8

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

x = (-b ± √(b^2 - 4ac)) / 2a

In this case, a = 1, b = 2, and c = -8. Plugging these values into the quadratic formula, we get:

x = (-(2) ± √((2)^2 - 4(1)(-8))) / 2(1) x = (-2 ± √(4 + 32)) / 2 x = (-2 ± √36) / 2 x = (-2 ± 6) / 2

Simplifying the expression, we get two possible values for x:

x = (-2 + 6) / 2 = 4 / 2 = 2 x = (-2 - 6) / 2 = -8 / 2 = -4

Therefore, the x-intercept(s) of the parabola y = x^2 + 2x - 8 are x = 2 and x = -4.

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 use the formula:

x = -b / 2a

In this case, a = 1 and b = 2. Plugging these values into the formula, we get:

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

To find the y-coordinate of the vertex, we need to plug the x-coordinate into the equation of the parabola:

y = x^2 + 2x - 8 y = (-1)^2 + 2(-1) - 8 y = 1 - 2 - 8 y = -9

Therefore, the coordinates of the vertex are (-1, -9).

Conclusion

In this article, we have found the x-intercept(s) and the coordinates of the vertex for the parabola y = x^2 + 2x - 8. The x-intercept(s) are x = 2 and x = -4, and the coordinates of the vertex are (-1, -9). These values can be used to graph the parabola and understand its behavior.

Graphing the Parabola

To graph the parabola, we can use the x-intercept(s) and the coordinates of the vertex. We can start by plotting the x-intercept(s) on the x-axis, and then draw a smooth curve through the vertex.

Here is a step-by-step guide to graphing the parabola:

  1. Plot the x-intercept(s) on the x-axis.
  2. Plot the vertex on the graph.
  3. Draw a smooth curve through the vertex, making sure to pass through the x-intercept(s).

By following these steps, we can create a graph of the parabola y = x^2 + 2x - 8.

Applications of the Parabola

The parabola y = x^2 + 2x - 8 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 mirror.

In addition, the parabola can be used to solve problems in physics, engineering, and economics. For example, it can be used to find the maximum or minimum value of a function, or to determine the optimal solution to a problem.

Conclusion

In conclusion, the parabola y = x^2 + 2x - 8 is a quadratic equation that can be used to model many real-world phenomena. By finding the x-intercept(s) and the coordinates of the vertex, we can gain a deeper understanding of the parabola and its behavior. The parabola has many real-world applications, and it can be used to solve problems in physics, engineering, and economics.

References

Note: The references provided are for educational purposes only and are not intended to be a comprehensive list of resources.

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 + 2x - 8. In this article, we will answer some frequently asked questions (FAQs) related to this topic.

Q: What is the x-intercept(s) of a parabola?

A: The x-intercept(s) of a parabola are the points where the parabola intersects the x-axis. In other words, it is the value(s) of x where the parabola crosses the x-axis.

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 a parabola?

A: The vertex of a parabola is the lowest or highest point on the parabola. It is the point where the parabola changes direction.

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 use the formula:

x = -b / 2a

where a and b are the coefficients of the quadratic equation.

Q: What is the significance of the x-intercept(s) and the vertex of a parabola?

A: The x-intercept(s) and the vertex of a parabola are important because they help us understand the behavior of the parabola. The x-intercept(s) tell us where the parabola intersects the x-axis, and the vertex tells us the lowest or highest point on the parabola.

Q: Can I use the x-intercept(s) and the vertex to graph a parabola?

A: Yes, you can use the x-intercept(s) and the vertex to graph a parabola. By plotting the x-intercept(s) on the x-axis and the vertex on the graph, you can draw a smooth curve through the vertex to create a graph of the parabola.

Q: What are some real-world applications of the parabola?

A: The parabola has many real-world applications, including modeling 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 mirror.

Q: Can I use the parabola to solve problems in physics, engineering, and economics?

A: Yes, you can use the parabola to solve problems in physics, engineering, and economics. For example, you can use the parabola to find the maximum or minimum value of a function, or to determine the optimal solution to a problem.

Q: What are some common mistakes to avoid when finding the x-intercept(s) and the vertex of a parabola?

A: Some common mistakes to avoid when finding the x-intercept(s) and the vertex of a parabola include:

  • Not setting y = 0 when finding the x-intercept(s)
  • Not using the correct formula to find the vertex
  • Not plotting the x-intercept(s) and the vertex correctly on the graph

Q: How can I practice finding the x-intercept(s) and the vertex of a parabola?

A: You can practice finding the x-intercept(s) and the vertex of a parabola by working through examples and exercises. You can also use online resources, such as graphing calculators or math software, to help you visualize the parabola and find the x-intercept(s) and the vertex.

Conclusion

In this article, we have answered some frequently asked questions (FAQs) related to finding the x-intercept(s) and the coordinates of the vertex for the parabola y = x^2 + 2x - 8. We hope that this article has been helpful in clarifying any confusion and providing a better understanding of the topic.

References

Note: The references provided are for educational purposes only and are not intended to be a comprehensive list of resources.