What Are The Coordinates Of The Vertex Of The Parabola With The Equation $y = X^2 + 2x - 3$?A. \[$(-1, -4)\$\] B. \[$(1, -4)\$\] C. \[$(3, -4)\$\] D. \[$(-3, 12)\$\]

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What are the coordinates of the vertex of the parabola with the equation y=x2+2x−3y = x^2 + 2x - 3?

Understanding the Problem

The problem requires finding the coordinates of the vertex of a parabola given by the equation y=x2+2x−3y = x^2 + 2x - 3. The vertex of a parabola is the point where the parabola changes direction, and it is represented by the coordinates (h, k). To find the coordinates of the vertex, we need to use the formula for the x-coordinate of the vertex, which is given by h=−b2ah = -\frac{b}{2a}, where a and b are the coefficients of the quadratic equation.

Finding the x-coordinate of the vertex

The given equation is y=x2+2x−3y = x^2 + 2x - 3. Comparing this equation with the standard form of a quadratic equation y=ax2+bx+cy = ax^2 + bx + c, we can see that a=1a = 1 and b=2b = 2. Now, we can use the formula for the x-coordinate of the vertex to find the value of h.

# Import necessary modules
import math

# Define variables
a = 1
b = 2

# Calculate the x-coordinate of the vertex
h = -b / (2 * a)

print("The x-coordinate of the vertex is:", h)

Finding the y-coordinate of the vertex

Now that we have the x-coordinate of the vertex, we can find the y-coordinate by substituting the value of h into the equation of the parabola.

# Define variables
x = h
a = 1
b = 2
c = -3

# Calculate the y-coordinate of the vertex
k = a * x**2 + b * x + c

print("The y-coordinate of the vertex is:", k)

Calculating the coordinates of the vertex

Now that we have the x and y-coordinates of the vertex, we can calculate the coordinates of the vertex.

# Define variables
h = -b / (2 * a)
k = a * h**2 + b * h + c

# Print the coordinates of the vertex
print("The coordinates of the vertex are:", (h, k))

Comparing the calculated coordinates with the given options

Now that we have the coordinates of the vertex, we can compare them with the given options to find the correct answer.

Option Coordinates
A (-1, -4)
B (1, -4)
C (3, -4)
D (-3, 12)

Conclusion

Based on the calculations, we can see that the coordinates of the vertex are (-1, -4). Therefore, the correct answer is:

The final answer is A. {(-1, -4)$}$
Vertex of a Parabola: A Comprehensive Guide

Understanding the Vertex of a Parabola

The vertex of a parabola is the point where the parabola changes direction. It is represented by the coordinates (h, k) and is a crucial concept in algebra and calculus. In this article, we will explore the concept of the vertex of a parabola, how to find its coordinates, and provide a comprehensive guide to help you understand this important topic.

Q&A: Vertex of a Parabola

Q: What is the vertex of a parabola?

A: The vertex of a parabola is the point where the parabola changes direction. It is represented by the coordinates (h, k) and is a crucial concept in algebra and calculus.

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

A: To find the coordinates of the vertex, you need to use the formula for the x-coordinate of the vertex, which is given by h=−b2ah = -\frac{b}{2a}, where a and b are the coefficients of the quadratic equation. Once you have the x-coordinate, you can find the y-coordinate by substituting the value of h into the equation of the parabola.

Q: What is the formula for the x-coordinate of the vertex?

A: The formula for the x-coordinate of the vertex is h=−b2ah = -\frac{b}{2a}, where a and b are the coefficients of the quadratic equation.

Q: How do I find the y-coordinate of the vertex?

A: To find the y-coordinate of the vertex, you need to substitute the value of h into the equation of the parabola. The equation of the parabola is given by y=ax2+bx+cy = ax^2 + bx + c, where a, b, and c are the coefficients of the quadratic equation.

Q: What is the significance of the vertex of a parabola?

A: The vertex of a parabola is significant because it represents the point where the parabola changes direction. It is also the minimum or maximum point of the parabola, depending on the direction of the parabola.

Q: How do I determine the direction of the parabola?

A: To determine the direction of the parabola, you need to look at the coefficient of the x^2 term in the equation of the parabola. If the coefficient is positive, the parabola opens upwards, and if the coefficient is negative, the parabola opens downwards.

Q: What is the relationship between the vertex and the axis of symmetry?

A: The vertex of a parabola is the point on the axis of symmetry. The axis of symmetry is a vertical line that passes through the vertex and is perpendicular to the x-axis.

Q: How do I find the axis of symmetry of a parabola?

A: To find the axis of symmetry of a parabola, you need to use the formula for the x-coordinate of the vertex, which is given by h=−b2ah = -\frac{b}{2a}. The axis of symmetry is a vertical line that passes through the vertex and is perpendicular to the x-axis.

Conclusion

In this article, we have explored the concept of the vertex of a parabola, how to find its coordinates, and provided a comprehensive guide to help you understand this important topic. We have also answered some frequently asked questions about the vertex of a parabola. We hope that this article has been helpful in understanding the concept of the vertex of a parabola.

Additional Resources

Vertex of a Parabola Formula

The formula for the x-coordinate of the vertex is h=−b2ah = -\frac{b}{2a}, where a and b are the coefficients of the quadratic equation.

Axis of Symmetry Formula

The axis of symmetry is a vertical line that passes through the vertex and is perpendicular to the x-axis. The formula for the x-coordinate of the vertex is h=−b2ah = -\frac{b}{2a}.

Vertex of a Parabola Examples

  • Find the coordinates of the vertex of the parabola y=x2+2x−3y = x^2 + 2x - 3.
  • Find the axis of symmetry of the parabola y=x2+2x−3y = x^2 + 2x - 3.

Answer Key

  • The coordinates of the vertex of the parabola y=x2+2x−3y = x^2 + 2x - 3 are (-1, -4).
  • The axis of symmetry of the parabola y=x2+2x−3y = x^2 + 2x - 3 is x = -1.