Math Problem Statement
Solution
The image contains three mathematical problems based on the function:
Let's break down and solve each of these parts:
(I) The problem asks: If the graph of is symmetric with respect to the line , find the value of .
- A quadratic function is symmetric about a vertical line when its vertex lies on that line. The vertex of a quadratic function is located at .
- In our case:
- The vertex must lie at .
Therefore:
Simplifying:
Multiply both sides by :
Expanding the right-hand side:
Solving for :
Thus, the value of is 1.
(II) Given , solve the inequality:
Substituting into the expression for :
So, we need to solve:
First, simplify the right-hand side:
This can be simplified by dividing the numerator by using polynomial long division, yielding:
Thus, the inequality becomes:
Now, solve this inequality.
(III) Solve for the values of such that for any , the inequality holds.
Substitute into the inequality:
Rearranging:
Simplifying:
This quadratic inequality needs to be solved to find the values of such that the inequality holds for .
Would you like further clarification or detailed solutions for any part? Here are 5 related questions to explore this topic further:
- How do you apply the vertex formula to find symmetry lines in quadratic functions?
- What is the process for solving quadratic inequalities?
- How do you perform polynomial long division?
- What are the conditions for a quadratic function to be positive over an interval?
- How does the value of affect the shape and position of the parabola defined by ?
Tip: When solving inequalities involving rational functions, always check for undefined points where the denominator equals zero.
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Math Problem Analysis
Mathematical Concepts
Quadratic Functions
Symmetry
Inequalities
Polynomial Division
Formulas
Quadratic function: f(x) = ax^2 + bx + c
Vertex formula: x = -b / (2a)
Polynomial division
Theorems
Properties of quadratic functions
Solving quadratic inequalities
Suitable Grade Level
Grades 10-12
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