Slope Line & Parabola

Drawing on the sources and our conversation history, we can discuss the slope of a line and compare it to the slope of a parabola (which is the graph of a quadratic function).

Slope of a Straight Line

The concept of the slope of a line is typically studied before moving on to quadratic functions. The slope of a line describes both its direction and steepness. In a general sense, the slope tells you how a small change in the x-direction affects the change in the y-direction. It is considered a measure of the rate of change.

There are several ways to define and calculate the slope of a line:

1. Rise by Run Ratio: Slope is understood as the ratio of the change in the y-direction (rise) to the change in the x-direction (run).
2. Using Two Points: Given two points (x₁, y₁) and (x₂, y₂) on the line, the slope (denoted by m) is calculated using the formula: m = (y₂ - y₁) / (x₂ - x₁).
3. Using Inclination: The slope can also be related to the angle of inclination (θ) that the line makes with the positive x-axis, where m = tan(θ).
   • For a horizontal line (parallel to the x-axis), the angle of inclination is 0 degrees, and the slope is 0. The equation of a horizontal line is y = a, where ‘a’ is a constant.
   • For a vertical line (parallel to the y-axis), the angle of inclination is 90 degrees. The tangent of 90 degrees is undefined, so the slope of a vertical line is undefined. The equation of a vertical line is x = b, where ‘b’ is a constant.

A key characteristic of a straight line is that its rate of change is constant. This means the slope is the same at every point along the line. The standard form of a linear function, f(x) = mx + c or y = mx + b, explicitly shows this constant slope ’m’ and the y-intercept ‘c’ or ‘b’.

Slope of a Parabola (Quadratic Function)

A quadratic function is described by the equation f(x) = ax² + bx + c, where ‘a’ is not equal to 0. The graph of any quadratic function is a parabola.

Unlike a straight line, the slope of a quadratic function is variable. It is not a constant value. The slope changes depending on the specific point on the parabola.

The sources describe a method to determine the slope of a quadratic function at any given point. This method generalises the concept of slope from straight lines. It involves calculating the slope of the line connecting two points on the curve that are symmetrically placed around the point of interest, say x₀. If you consider points x₀ - h and x₀ + h, the slope is calculated as the change in y divided by the change in x between these two points:

Slope = (f(x₀ + h) - f(x₀ - h)) / ((x₀ + h) - (x₀ - h))

The denominator simplifies to 2h. By substituting f(x) = ax² + bx + c into the numerator and simplifying the expression (as shown for y = x² and the general case), the slope of the quadratic function f(x) = ax² + bx + c at any point x (or x₀) is found to be 2ax + b (or 2ax₀ + b). This gives a new function, often denoted g(x), that represents the slope at any x.

The slope formula, 2ax + b, depends on the coefficients ‘a’ and ‘b’, but it does not depend on the constant term ‘c’.

A significant point related to the slope of a quadratic function is where the slope is equal to zero. Setting the slope expression to zero, 2ax + b = 0, and solving for x gives x = -b/(2a). This x-value corresponds precisely to the x-coordinate of the vertex of the parabola. At the vertex, the rate of change of the function with respect to x becomes zero. This signifies the point where the function transitions from increasing to decreasing (if the parabola opens downwards, a < 0) or from decreasing to increasing (if the parabola opens upwards, a > 0). Therefore, the point where the slope is zero marks the location where the quadratic function reaches its minimum or maximum value.