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:

f(x)=1.8x216.1x38.5f(x) = -1.8x^2 - 16.1x - 38.5

we start by noting that this is a parabola that opens downward (since the coefficient of x2x^2 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 ax2+bx+cax^2 + bx + c is:

x=b2ax = -\frac{b}{2a}

For the function f(x)=1.8x216.1x38.5f(x) = -1.8x^2 - 16.1x - 38.5, we have:

  • a=1.8a = -1.8
  • b=16.1b = -16.1

Now, let's calculate the x-coordinate of the vertex:

x=16.12(1.8)=16.13.6x = -\frac{-16.1}{2(-1.8)} = \frac{16.1}{3.6}

I'll calculate this value for you.The x-coordinate of the vertex is approximately x=4.47x = -4.47.

Next, to find the maximum value of the function, substitute this xx-value into the original equation:

f(x)=1.8x216.1x38.5f(x) = -1.8x^2 - 16.1x - 38.5

I'll calculate the maximum value f(4.47)f(-4.47).The maximum value of the function is approximately f(x)=2.50f(x) = -2.50 to the nearest hundredth.

Do you have any questions or would you like further details?

Here are 5 related questions:

  1. How do we determine whether a quadratic function has a maximum or minimum?
  2. What is the significance of the vertex in a quadratic function?
  3. How do the coefficients aa, bb, and cc in a quadratic function affect its graph?
  4. Can a parabola have more than one maximum or minimum point?
  5. What other methods can be used to find the maximum value of a quadratic function?

Tip: Always check the sign of the x2x^2 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