Understanding How to Draw a Tangent on a Graph for 14-Year-Old Students

What is a Tangent?

A tangent is a straight line that touches a curve at exactly one point without crossing it at that point. This means the tangent line just 'touches' the curve and has the same slope as the curve does at that point.

How to Draw a Tangent Line on a Graph

1. Identify the Point on the Curve: First, find the exact point on the graph where you need to draw the tangent. This point might be given or you may need to find it by looking at the graph or the function.
2. Understand the Slope at the Point: The slope of the tangent line at the point is the same as the slope (rate of change) of the curve at that point.
3. Calculate or Estimate the Slope: - If you have the equation of the curve (for example, y = x²), find the derivative (dy/dx), which gives the slope of the curve at any point.
   - If you don’t know derivatives yet, estimate the slope by drawing a line that just touches the curve at your point and is as close as possible to the curve near that point.
4. Draw the Tangent Line: Using the point you found and the slope you calculated or estimated, draw a straight line that touches the curve only at that point and matches that slope.

Example:

Let’s say the curve is y = x² and you want to draw a tangent at x = 1:

• Find y at x=1: y = 1² = 1, so the point is (1,1).
• Find the slope: The derivative dy/dx = 2x, so at x = 1, slope = 2.
• Equation of the tangent line at (1,1) is y - 1 = 2(x - 1), simplifying to y = 2x - 1.
• Draw this straight line on your graph, and it will just touch the curve y = x² at point (1,1).

By following these steps, you can successfully draw a tangent line on any graph!