Slope of quadratic function

Drawing upon the provided sources and our conversation history, the slope of a quadratic function is a key concept discussed, contrasting it with the constant slope of a linear function.

Here’s what the sources say about the slope of a quadratic function:

1. Contrast with Linear Functions:

   • For a linear function y = mx + c (or f(x) = mx + c), the quantity ’m’ represents the slope, and it is constant. This slope represents the sensitivity of the expression to changes in the x-direction, indicating how a small change in x affects the change in y. It is calculated as the ratio of change in y to change in x (rise by run) and can be related to the tangent of the angle of inclination with the positive x-axis.
   • Unlike linear functions, the slope of a quadratic function is variable.

2. Definition and Formula:

   • For a quadratic function f(x) = ax² + bx + c where a is not equal to 0, the slope of the function at any given point x is given by the expression 2ax + b. This formula for the slope is denoted as g(x) = 2ax + b.
   • The slope depends on the coefficients ‘a’ and ‘b’, but notably, it does not depend on the constant term ‘c’.

3. Derivation/Calculation Method:

   • The sources explain a method to determine this slope, which involves considering two points symmetrically placed around a point x₀ on the curve. If you take points x₀ - h and x₀ + h, the slope at x₀ is approached by the slope of the line connecting the points (x₀ - h, f(x₀ - h)) and (x₀ + h, f(x₀ + h)).
   • Using the formula (f(x₀ + h) - f(x₀ - h)) / ((x₀ + h) - (x₀ - h)), which simplifies to (f(x₀ + h) - f(x₀ - h)) / (2h), and substituting f(x) = ax² + bx + c, the calculation leads to the slope being 2ax₀ + b at the point x₀. This method can be generalized to a general quadratic function.

4. Significance of Zero Slope:

   • An important observation is made when the slope is equal to 0. Setting the slope expression 2ax + b equal to 0 and solving for x gives x = -b/(2a).
   • This specific x-value, x = -b/(2a), is the x-coordinate of the vertex of the parabola.
   • The reason the slope is 0 at the vertex is that the rate of change of the function with respect to x becomes zero at this point. The function is changing direction, either switching from increasing to decreasing (at a maximum) or from decreasing to increasing (at a minimum).
   • Therefore, the point where the slope is 0 corresponds to the x-value where the quadratic function reaches its minimum or maximum value.

5. Examples:

   • For the standard prototype example y = x², where a=1, b=0, c=0, the slope at any point x (or x₀) is calculated as 2x (or 2x₀). At x=0, the slope is 0, corresponding to the vertex (0,0).
   • For the function f(x) = x² + 4x + 4, where a=1 and b=4, the slope is 2x + 4. Setting the slope to 0 (2x + 4 = 0) gives x = -2, which is the x-coordinate of the axis of symmetry and the vertex.

In summary, the slope of a quadratic function is a variable quantity described by the linear expression 2ax + b, which is derived using a method that generalizes the slope concept from straight lines. The point where this slope is zero is particularly significant as it corresponds to the vertex, where the function attains its minimum or maximum value.