Math Problem Statement
Solution
The quadratic function in question is . Let's analyze the key properties:
1. Axis of Symmetry (A.O.S.)
The axis of symmetry for a quadratic function in the form is given by: For , and . Plugging these values into the formula: So, the A.O.S. is .
2. Vertex
The vertex is on the axis of symmetry, and its x-coordinate is . To find the y-coordinate, substitute into the function: Thus, the vertex is at .
3. Opens
Since the coefficient of (which is ) is negative, the parabola opens downwards.
4. Max/Min Value
Because the parabola opens downwards, the vertex represents the maximum value of the function, which is .
5. Domain
The domain of any quadratic function is always all real numbers: .
6. Range
Since the function has a maximum value at the vertex, the range is .
7. y-Intercept
The y-intercept occurs where . Substituting into the function: Thus, the y-intercept is .
8. Solutions
To find the solutions (or x-intercepts), solve the quadratic equation using the quadratic formula: For , , and : Simplifying further: Breaking it down: Thus, the solutions are approximately and .
Let me know if you'd like further details on any of these points or additional explanations!
Relative Questions:
- What is the significance of the axis of symmetry in a quadratic function?
- How do you determine if a parabola opens upwards or downwards?
- Why is the domain of a quadratic function always all real numbers?
- How does the vertex relate to the maximum or minimum value of the function?
- What role does the discriminant play in determining the number of real solutions?
Tip:
When solving quadratic equations, always check the sign of the leading coefficient to know whether the parabola opens upwards or downwards.
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Math Problem Analysis
Mathematical Concepts
Algebra
Quadratic Functions
Vertex Form
Parabolas
Formulas
Axis of symmetry formula: x = -b / (2a)
Vertex form of a quadratic function
Quadratic formula: x = (-b ± √(b² - 4ac)) / 2a
Theorems
Properties of quadratic functions
Suitable Grade Level
Grades 9-10
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