Math Problem Statement

the parabola has two zeros, opens downward and is wider than y=x^2 and the axis of symmetry is x=4

Solution

To describe a parabola with these characteristics, we can start with the general equation of a parabola:

y=a(xh)2+ky = a(x - h)^2 + k

Given Information:

  1. Two zeros: This indicates the parabola crosses the x-axis at two points.
  2. Opens downward: This means a<0a < 0 (negative leading coefficient).
  3. Wider than y=x2y = x^2: This indicates a<1|a| < 1.
  4. Axis of symmetry x=4x = 4: The vertex is on this line, so h=4h = 4.

Constructing the Equation:

  1. The vertex is at (4,k)(4, k), but the value of kk is not directly given.
  2. We assume the zeros are equidistant from the vertex, and the parabola opens downward.

Let the zeros be (p,0)(p, 0) and (q,0)(q, 0), and their midpoint will be the vertex. Hence:

p + q = 8 \quad \text{(since the axis of symmetry is x = 4)}.

Let’s write a possible equation based on these properties: y=a(x4)2+ky = a(x - 4)^2 + k where a<0a < 0 and a<1|a| < 1.

If you have specific zeros or additional information (such as the yy-intercept), we can refine this equation.


Follow-Up Questions:

  1. Do you know the specific zeros of the parabola?
  2. Can you provide the vertex's yy-coordinate (kk) or another point on the parabola?
  3. Should we assume a specific value of aa to explore further details?
  4. Do you need the parabola to pass through a specific point?
  5. Would you like to graph this parabola based on these properties?

Tip:

When a parabola is "wider," the coefficient aa controls the "stretch." A smaller a|a| results in a wider parabola.

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Math Problem Analysis

Mathematical Concepts

Quadratic Functions
Vertex Form of a Parabola
Parabola Properties

Formulas

Vertex form of a parabola: y = a(x - h)^2 + k
Axis of symmetry: x = -b / (2a)

Theorems

Properties of Parabolas

Suitable Grade Level

Grades 9-11