Math Problem Statement
f(x) = 4x²-2x+12
Solution
The function given is:
Let's break down its key features:
1. Type of Function:
This is a quadratic function, which means its graph will be a parabola. Since the coefficient of is positive (4), the parabola opens upwards.
2. Vertex:
The vertex form of a quadratic function can be found using the vertex formula:
For , we have:
So, the x-coordinate of the vertex is:
Now, plug this value back into the function to find the y-coordinate of the vertex:
So, the vertex is at:
3. Axis of Symmetry:
The axis of symmetry is the vertical line that passes through the vertex, given by:
4. Y-intercept:
To find the y-intercept, set :
So, the y-intercept is at .
5. Range and Domain:
- Domain: The domain of any quadratic function is all real numbers, so .
- Range: Since the parabola opens upwards and the vertex is the minimum point, the range is .
Summary:
- Vertex:
- Axis of Symmetry:
- Y-intercept:
- Domain:
- Range:
Would you like more details or have any questions?
Related Questions:
- How do you find the x-intercepts of ?
- What is the significance of the discriminant in a quadratic function?
- How can we convert into vertex form?
- What are the properties of the graph of ?
- How does changing the coefficient of affect the shape of the parabola?
Tip: The discriminant of a quadratic equation, , helps determine the nature of its roots (real or complex).
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Math Problem Analysis
Mathematical Concepts
Algebra
Quadratic Functions
Parabolas
Formulas
Vertex formula: x = -b / 2a
Quadratic function standard form: ax² + bx + c
Theorems
Properties of quadratic functions
Axis of symmetry
Vertex
Suitable Grade Level
Grades 9-10
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