What equation represents a parabola with a vertex at the point (5, 3)?

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To determine which equation has a graph that is a parabola with a vertex at the point (5, 3), we can use the vertex form of a quadratic equation. The vertex form is expressed as:

y = a(x - h)² + k

In this equation:

  • (h, k) are the coordinates of the vertex.
  • a determines the direction and width of the parabola.

Given the vertex (5, 3), we can substitute h = 5 and k = 3 into the vertex form:

y = a(x - 5)² + 3

Now, the value of a will influence the shape and orientation of the parabola:

  • If a > 0, the parabola opens upwards.
  • If a < 0, the parabola opens downwards.

For example:

  • If we let a = 1, the equation becomes:
    • y = (x - 5)² + 3
  • If we let a = -1, the equation becomes:
    • y = -(x - 5)² + 3

Both of these equations represent parabolas with a vertex at (5, 3); one opens upwards while the other opens downwards. Therefore, to form a parabola with a vertex at (5, 3), you can use the equation:

y = a(x - 5)² + 3

where a can be any non-zero real number, allowing you to adjust the orientation and steepness of the parabola while keeping the vertex fixed at (5, 3).

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