Math Problem Statement
Find the maximum value of the function f, of, x, equals, minus, 1, point, 8, x, squared, minus, 16, point, 1, x, minus, 38, point, 5f(x)=−1.8x 2 −16.1x−38.5 to the nearest hundredth.
Solution
To find the maximum value of the quadratic function:
we start by noting that this is a parabola that opens downward (since the coefficient of is negative), so it has a maximum point, which occurs at the vertex.
The formula for the x-coordinate of the vertex for a quadratic function is:
For the function , we have:
Now, let's calculate the x-coordinate of the vertex:
I'll calculate this value for you.The x-coordinate of the vertex is approximately .
Next, to find the maximum value of the function, substitute this -value into the original equation:
I'll calculate the maximum value .The maximum value of the function is approximately to the nearest hundredth.
Do you have any questions or would you like further details?
Here are 5 related questions:
- How do we determine whether a quadratic function has a maximum or minimum?
- What is the significance of the vertex in a quadratic function?
- How do the coefficients , , and in a quadratic function affect its graph?
- Can a parabola have more than one maximum or minimum point?
- What other methods can be used to find the maximum value of a quadratic function?
Tip: Always check the sign of the coefficient to determine whether a quadratic opens upwards (minimum) or downwards (maximum).
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Math Problem Analysis
Mathematical Concepts
Algebra
Quadratic Functions
Vertex of a Parabola
Formulas
Vertex formula: x = -b / 2a
Quadratic function: f(x) = ax^2 + bx + c
Theorems
The maximum or minimum value of a quadratic function occurs at its vertex.
Suitable Grade Level
Grades 9-10
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