Math Problem Statement
Solution
The problem provided is about a function , where . There are three parts to the problem:
- Part a: Compute the function for specific values of , i.e., , , , , , and .
- Part b: Plot the graph of the function.
- Part c: Given the domain , determine the range of the function over this domain.
Part a: Calculate for the given values of
The function is . Let's compute:
Part b: Graph of the function
The function is a quadratic equation , which forms a parabola. The vertex of the parabola can be found using the formula for the vertex of a quadratic function, , where and .
- Substituting into the function: .
Thus, the vertex of the parabola is , and the parabola opens upwards (since the coefficient of is positive). You can sketch the parabola based on this information, noting the symmetry around .
Part c: Range of the function
The domain is given as . To find the range, we evaluate the function at the boundaries of the domain and at the vertex (since it's a quadratic function):
- We already computed , , and .
- The minimum value of occurs at the vertex, so the range of the function is .
Summary of Answers:
- , , , , ,
- The graph is a parabola with a vertex at and it opens upwards.
- The range is .
Would you like further details or clarifications?
Here are 5 additional related questions:
- How do you find the axis of symmetry for a quadratic function?
- How can you determine if a parabola opens upwards or downwards?
- What is the significance of the vertex in the graph of a quadratic function?
- How do you find the roots of a quadratic equation like ?
- How does the domain affect the range in quadratic functions?
Tip: For any quadratic function , the vertex represents the maximum or minimum point, depending on whether is negative (maximum) or positive (minimum).
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Math Problem Analysis
Mathematical Concepts
Algebra
Quadratic Functions
Graphing
Formulas
f(x) = ax^2 + bx + c
x_vertex = -b / 2a
Theorems
Quadratic Formula
Suitable Grade Level
Grades 9-11
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