Math Problem Statement
- The figure shows the graph of the quadratic function y = x2 + bx + c, y where b and c are constants. (Given that have two pt (0,5), (6,5)), the graph opens upward) (a) Find the values of b and c. (b) Find the minimum value of y.
Solution
To solve the given problem, we need to find the constants and of the quadratic function , given the points and .
Part (a): Find the values of and
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Using the point :
Since lies on the graph, we can substitute and into the equation:
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Using the point :
Since also lies on the graph, we substitute and :
Therefore, the values of the constants are:
Part (b): Find the minimum value of
The minimum value of for a quadratic function (where ) occurs at the vertex of the parabola. The x-coordinate of the vertex can be found using the formula:
For our quadratic function , and :
To find the minimum value of , we substitute back into the quadratic equation:
Therefore, the minimum value of is:
Do you want any further details or have any questions?
Here are some questions you might want to ask next:
- How do you find the vertex of a quadratic function?
- What is the significance of the vertex in determining the minimum or maximum value of a quadratic function?
- How can you determine if a quadratic function opens upwards or downwards?
- What are some methods to solve quadratic equations?
- How does the value of the discriminant affect the nature of the roots of a quadratic equation?
Tip: When working with quadratic functions, identifying key points such as the vertex and the intercepts can provide valuable insights into the graph and help solve related problems effectively.
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Math Problem Analysis
Mathematical Concepts
Quadratic Functions
Vertex of a Parabola
Minimum Value of Quadratic Function
Formulas
Quadratic equation formula: y = ax^2 + bx + c
Vertex formula: x = -b / (2a)
Minimum value formula: y = f(-b / (2a))
Theorems
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Suitable Grade Level
High School
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