Introduction to Normal Line Calculator

Normal line equation calculator is an online tool used to calculate the equation of the normal line. It uses the normal line equation and the slope formula to calculate the equation. In other words, it is used to calculate the line perpendicular to the tangent line at the given point.

In mathematics, the derivative has many applications such that it can be used to find the tangent line of a function and its slope. So we introduce a tool online that can handle any function to calculate its normal line.

Formula used by Normal Line Formula Calculator

This calculator is used to find the normal line equation using tangent at a given point. It uses following step-by-step formulas to calculate the equation:

  • It calculates the value of the function at the given point by putting:
  • $$ x \;=\; x_o $$


    $$ f(x_o) \;=\; y_o $$
  • It calculates the slope M of the normal line at:
  • $$ x \;=\; x_o $$

    by using this formula:

    $$ M(x_o) \;=\; - \frac{1}{f'(x_o)} $$
  • The general form of the equation of tangent passing through:
  • $$ (x_o \;,\; y_o) $$

    And having slope M is:

    $$ (y \;-\; y_o) \;=\; m(x \;-\; x_o) $$

The normal line calculator uses this formula to calculate normal line equation using slope M, given point pf:

$$ x\;=\; x_o $$

And the value of function at the given point of:

$$ y_o $$

How to use the Calculator to Find the Normal Line?

Using this tool, the tangent line and the equation of a normal line can be formed. You need to follow some steps to use it. These steps are given below:

  1. In the first step, enter the value of the function in the “Enter Function” box, or you can use the “Load Example” option to try the solution.
  2. Now enter the point of x in the “At the point of x” box.
  3. You can review the function to check the values. Then click on the “Calculate” button.

You will get the equation or normal line a few seconds after clicking the calculate button.

Why use a Normal Line Equation Calculator?

The tangent is the straight line that touches the given curve at a given point. And the normal line is perpendicular to the tangent. So we can say that the normal line is associated with the deviation at a given point. While computing the normal line equation, students may get stuck with this method. Or they may forget the point-slope form of the equation. It would be best to use the normal line finder because it provides you with the step-by-step process and the graph as a visual representation of the given curve.

Benefits of Calculating Normal Line

In geometry, you can only be an expert in calculations if you practice. The normal line calculator equation helps you practice with many different curves. Some other benefits of this tool are:

  1. It clarifies the result with the graph by showing where the point x touches the given curve and the tangent to the curve.
  2. It is reliable because it never gives false results.
  3. It can save your time by giving the result with every defined step.
  4. You can practice with unlimited examples for free by using this tool to find the equation of a normal line.
  5. Normal line equation calculator can handle any curve like parabola, ellipse, and hyperbola, making it more beneficial for students.

How to Find the Equation of a Normal Line?

Consider a curve given as:

$$ y \;=\; 4x^2 $$

We have to find the normal line at the point of:

$$ x_o \;=\; 1 $$

So, the value of the given function at this point is,

$$ y_o \;=\; 4(1)^2 \;=\; 4 $$

And the slope can be found by,

$$ M(x_o) \;=\; - \frac{1}{f'(x_o)} $$


$$ f'(x_o) \;=\; 8 $$


$$ M(x_o) \;=\; - \frac{1}{8} $$

The point slope form of the equation is,

$$ (y \;-\; y_o) \;=\; M(x \;-\; x_o) $$

So, the equation of normal line at (1, 4) can be calculated as,

$$ (y \;-\; 4) \;=\; -\; \frac{1}{8} (x \;-\; 1) $$ $$ y \;=\; -\; \frac{x}{8} \;+\; \frac{1}{8} \;+\; 4 $$ $$ y \;=\; -\; \frac{x}{8} \;+\; \frac{33}{8} $$

We are sure that you’ll like this normal line calculator at a point.

Alan Walker

Alan Walker

Last Updated June 02, 2022

Studies mathematics sciences, and Technology. Tech geek and a content writer. Wikipedia addict who wants to know everything. Loves traveling, nature, reading. Math and Technology have done their part, and now it's the time for us to get benefits.