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Stationary points are involved in both physics and mathematics, particularly in the field of calculus. As the word stationary means stillness or not moving, the same is implied for this. The Stationary point of a specific variable under study is the point whose function’s derivative is zero. According to Layman’s statement, it is a point where the function of the variable can neither increase nor decrease.

How to find the slope of a graph?

The slope of a line is the value that describes the steepness of the line and is usually determined by calculating the vertical distance to horizontal distance. In mathematics, we mostly calculate the steepness of a given line, that is, slope.

The process of finding the slope of a line is to use the formula rise/run. So, if we are given a slope of a line on a graph first, we need to find the equation. Choose any two points like (x1, y1) and (x2, y2).

Thus, the formula for calculating slope will be,

Finding the slope of a graph- 

  1. Select two random points that are integer coordinates from the graph.
  2. Label them with any codes, for example- A and B
  3. Calculate the rising value that is vertically (Y-axis) from A to B; if the value is in the upward direction, then it is positive and downward, then negative.
  4. Calculate the running value that is horizontally (X-axis) from A to B; towards the right, the value will be positive, and for the left, it will be negative.
  5. Then use the formula rise/run to obtain the slope of the graph given.
  • Using slope formula-
  1. Select any two points on the line from the graph.
  2. The slope of the line represents two points like (x1, y1) and (x2, y2)
  3. Then, apply the formula, where m is the slope.

m = (y2-y1 )/(x2-x1)

Stationary Points-

The stationary point is also called the stagnant point, and for a differential equation, this stationary point on the graph is where the curve’s gradient signifies zero. The stationary line can be used to determine the tangential line on the graph because the stationary point on a curve is the point at which the tangent line is either horizontal or vertical. It is also called the critical point.

The location of the stationary curve is employed in curve sketching.

If       

               y = f (x)

then the stationary point will be

 

f ‘(x)=0

or   

 dy/dx = 0

The slope of the tangent formula-

The concept for tangents and lines must be clear to understand the tangent formula for slope. The slope is the ratio of the difference in the y coordinates divided by the differences of x coordinates and represented as,

m = (y2-y1 )/(x2-x1)

Here, the tanθ is the same as the slope (m), and it can be negative or positive depending on the value taken. The slopes in the parallel lines are the same, and those of perpendicular lines are -1.

The tangent line is the one that touches a curve at a point on the graph.

Finding the slope of the tangent line-

The slope can be calculated by finding the derivative of the curve that is

f (x and finding the derivative at the point where the tangent and curve meet)

Thus, the formula for the slope of the tangent line is given by if points are (x1, y1)

y-y1 = m (x-x1)

y = MX + c

For example, if f (x)= x2 at point (1,2)

dy/dx=d(x²)/dx

         = 2x

       f’(x)= d(x²)/dx

    = 2(1)

      = 2

By tangent formula, 

y-2 = 2(x-1)

  y-2+2= 2x

y = 2x

Conclusion

The stationary points and the slope of a line are a part of calculus and trigonometric functions. The practical application of stationary points is generally used in calculating the minimum and maximum points on the graph. In a graph that consists of one or two variables, the stationary point can be easily calculated. As for the slope, in mathematics, the slope of a line on a graph is often calculated. There are multiple methods for calculating the slope, which two ways have been discussed here.

The tangent formula of the slope is said to be the same as the value of m. Depending on the movement of the line, like upwards and downwards, the slope can be positive and negative. Therefore, the slope aspect is very important in the landscape metric, geographical information and analysing of the elevation data.