Math Problem Statement
Use graphing technology to find the range of the function f, of, x, equals, x, squared, plus, 6, x, plus, 1, .f(x)=x^2+6x+1.
Solution
To find the range of the function using graphing technology, let's first analyze the function and then describe the graph.
Analyzing the Function
The function is a quadratic function in standard form , where , , and .
Vertex of the Parabola
The vertex form of a quadratic function can be derived by completing the square or using the vertex formula:
To find the corresponding value at this :
So, the vertex of the parabola is at .
Graphing the Function
The function is a parabola that opens upwards (since the coefficient of is positive). The vertex is at , which is the minimum point on the graph.
Determining the Range
Since the parabola opens upwards and the lowest point is at , the range of is all values of that are greater than or equal to .
Conclusion
Thus, the range of the function is:
Would you like more details on the graph or have any questions?
Here are some related questions:
- What is the domain of the function ?
- How does changing the coefficient of in a quadratic function affect the range?
- What is the range of the function ?
- How can the vertex form of a quadratic function help in determining the range?
- What transformations can be applied to the function to obtain ?
Tip: The vertex of a parabola gives you the minimum or maximum value of the quadratic function, which is key to determining the range.
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Math Problem Analysis
Mathematical Concepts
Quadratic Functions
Vertex Form
Graphing
Formulas
Vertex formula for a quadratic function
Theorems
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Suitable Grade Level
Grades 9-12