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:
Given Information:
- Two zeros: This indicates the parabola crosses the x-axis at two points.
- Opens downward: This means (negative leading coefficient).
- Wider than : This indicates .
- Axis of symmetry : The vertex is on this line, so .
Constructing the Equation:
- The vertex is at , but the value of is not directly given.
- We assume the zeros are equidistant from the vertex, and the parabola opens downward.
Let the zeros be and , 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: where and .
If you have specific zeros or additional information (such as the -intercept), we can refine this equation.
Follow-Up Questions:
- Do you know the specific zeros of the parabola?
- Can you provide the vertex's -coordinate () or another point on the parabola?
- Should we assume a specific value of to explore further details?
- Do you need the parabola to pass through a specific point?
- Would you like to graph this parabola based on these properties?
Tip:
When a parabola is "wider," the coefficient controls the "stretch." A smaller 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
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