{Use of Tech} Fixed points of quadratics and quartics Let f(x) = ax(1 -x), where a is a real number and 0 ≤ a ≤ 1. Recall that the fixed point of a function is a value of x such that f(x) = x (Exercises 48–51).​​ 


c. Graph g for a = 2, 3, and 4.

Verified step by step guidance

1. Understand the concept of a fixed point: A fixed point of a function f(x) is a value x such that f(x) = x. This means that when you substitute x into the function, the output is the same as the input.
2. For the function f(x) = ax(1 - x), we need to find the fixed points by solving the equation f(x) = x. This translates to ax(1 - x) = x.
3. Rearrange the equation ax(1 - x) = x to find the values of x that satisfy it. Start by expanding the left side: ax - ax^2 = x.
4. Move all terms to one side to form a quadratic equation: ax - ax^2 - x = 0, which simplifies to (a - 1)x - ax^2 = 0.
5. Factor the quadratic equation or use the quadratic formula to find the values of x. The solutions to this equation will be the fixed points of the function for the given values of a.

Key Concepts

Fixed Points

A fixed point of a function is a value of x where the function's output equals its input, i.e., f(x) = x. This concept is crucial for understanding the behavior of functions, as fixed points can indicate stable or unstable equilibria. In the context of the given function f(x) = ax(1 - x), finding fixed points involves solving the equation ax(1 - x) = x.

Quadratic Functions

Quadratic functions are polynomial functions of degree two, typically expressed in the form f(x) = ax^2 + bx + c. The function provided, f(x) = ax(1 - x), can be rewritten as f(x) = -ax^2 + ax, which is a quadratic function. Understanding the properties of quadratics, such as their parabolas' shapes and vertex locations, is essential for graphing and analyzing the function's behavior.

Graphing Techniques

Graphing techniques involve plotting points and understanding the shape and behavior of functions on a coordinate plane. For the function f(x) = ax(1 - x), varying the parameter 'a' affects the width and position of the parabola. When graphing for different values of 'a' (2, 3, and 4), one must consider how these changes influence the fixed points and overall graph shape, which is vital for visualizing the function's dynamics.