Math Problem Statement

f(x)=–5(x–8)2.

Solution

The given function is:

f(x)=5(x8)2f(x) = -5(x - 8)^2

Analysis of the Function

  1. General Form: This is a quadratic function in vertex form: f(x)=a(xh)2+kf(x) = a(x - h)^2 + k Here, a=5a = -5, h=8h = 8, and k=0k = 0.

  2. Vertex: The vertex of the parabola is at (h,k)=(8,0)(h, k) = (8, 0).

  3. Direction of Opening: Since a=5a = -5, the parabola opens downward because the coefficient of the squared term is negative.

  4. Steepness: The absolute value of aa (here, a=5|a| = 5) determines the steepness of the parabola. A larger value of a|a| indicates a steeper graph.

  5. Axis of Symmetry: The axis of symmetry is the vertical line passing through the vertex, given by: x=h=8x = h = 8

  6. Maximum Value: Since the parabola opens downward, the vertex represents the maximum value of the function: f(8)=5(88)2=0f(8) = -5(8 - 8)^2 = 0 Thus, the maximum value is 00 at x=8x = 8.

Key Points:

  1. Vertex: (8,0)(8, 0)
  2. Axis of Symmetry: x=8x = 8
  3. Direction: Downward
  4. Maximum Value: f(x)=0f(x) = 0
  5. Domain and Range:
    • Domain: (,)(-\infty, \infty)
    • Range: (,0](-\infty, 0]

Would you like a graph or further details about this function?

Related Questions:

  1. What is the y-intercept of the function f(x)=5(x8)2f(x) = -5(x - 8)^2?
  2. How do we calculate specific values of f(x)f(x) for given xx-values?
  3. How does changing the value of aa affect the parabola’s shape?
  4. What is the effect of shifting hh and kk in the vertex form of a quadratic equation?
  5. Can this function be rewritten in standard form ax2+bx+cax^2 + bx + c? If yes, how?

Tip:

When analyzing quadratic functions, always identify the vertex and axis of symmetry first—they are key to understanding the function's behavior.

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Math Problem Analysis

Mathematical Concepts

Quadratic Functions
Vertex Form of a Quadratic Equation
Parabolas
Domain and Range

Formulas

f(x) = a(x - h)^2 + k

Theorems

Vertex form of quadratic equations
Properties of parabolas

Suitable Grade Level

Grades 8-10