Dynamic Calculus

Home > The gradient function

Tangents and normals

The graph of the parabola \(y = {x^2}\) is drawn below. What are the equations of the tangent and normal to the curve at the point \(P\)?
[You can reset the applet at any time by clicking the 'circular arrows' icon in the top right corner]

 

 

  1. On grid paper, draw an accurate sketch of the parabola \(y = {x^2}\). Draw the tangent to the curve at the point \((-4, 16)\) and measure the gradient.
  2. Using calculus, find the gradient of the tangent to the curve at the point \((-4, 16)\). How does your answer from question (1) compare?
  3. Using the 'point-gradient' formula (or otherwise), write down the equation of the tangent at the point \((-4, 16)\). Check your solution by clicking the 'show equation and graph of tangent' checkbox.
  4. What is the geometric relationship between the tangent and the normal to a curve at a point? On your sketch, draw the normal to the curve at the point \((-4, 16)\).
  5. Using the 'point-gradient' formula (or otherwise), write down the equation of the normal at the point \((-4, 16)\). Check your solution by clicking the 'show equation and graph of normal' checkbox.
  6. Reposition the point \(P\) by moving the 'slider' left and right. What is the location of the point \(P\) if the normal has equation \(y = \frac{x}{2} + 1.5\)?