Understanding the Equation of a Parabola

The equation of a parabola can be determined using its key features such as the vertex (turning point) and x-intercepts (roots). This article will guide you through the process of finding the equation of a parabola given a specific turning point and x-intercepts. We will explore the key concepts, common mistakes, and provide a step-by-step solution.

Key Concepts

Vertex (Turning Point): The vertex of a parabola represents its maximum or minimum point, depending on whether the parabola opens upward or downward. X-Intercepts: The points where the parabola crosses the x-axis, representing the roots or solutions to the quadratic equation. Axis of Symmetry: The vertical line that passes through the vertex, dividing the parabola into two symmetric halves.

Correct Terminology and Common Mistakes

Your question mentions a "turning point". In mathematical terms, this is typically referred to as the vertex. Additionally, it is crucial to ensure that the given information is consistent. For instance, if the vertex is at -15, the x-intercepts should be symmetric with respect to the axis of symmetry.

The Given Problem

Your question states that the vertex of the parabola is at -15 and the x-intercepts are at -3 and -1. However, these values are inconsistent. The vertex must lie on the axis of symmetry, which is equidistant from both x-intercepts. Therefore, if the x-intercepts are -3 and -1, the axis of symmetry would be at x -2. Consequently, the vertex should be at -2, not -15.

Revised Problem and Solution

Let's assume the vertex is at -2 and the x-intercepts are -3 and -1. We can use the vertex form of the parabola equation:

Vertex Form: (y a(x - h)^2 k), where (h, k) is the vertex.

Here, h -2 and k -15. Substituting these values, we get:

Step 1: Write the vertex form equation:

(y a(x 2)^2 - 15)

Step 2: Use one of the x-intercepts to find the value of a. Since the x-intercepts are -3 and -1, we can use -1:

0 a(-1 2)^2 - 15

0 a(1) - 15

15 a

Step 3: Substitute a back into the equation:

(y 15(x 2)^2 - 15)

This is the equation of the parabola with the given vertex and x-intercepts.

Summary

In summary, to find the equation of a parabola given a specific turning point and x-intercepts, follow these steps:

Ensure the vertex is on the axis of symmetry, which is midway between the x-intercepts. Use the vertex form of the parabola equation: (y a(x - h)^2 k). Substitute the vertex coordinates (h, k) into the equation. Use one of the x-intercepts to solve for the coefficient a. Substitute the value of a back into the equation.

This process ensures that your equation accurately represents the given conditions.

Key Takeaways

The vertex of a parabola is critical in determining the equation. Consistency in the given information is crucial. The axis of symmetry is half-way between the x-intercepts.

By following these steps and understanding the key concepts, you can solve problems involving parabolas with confidence.