Lesson 5: Tangents and Gradients of Curves

👨🏽‍🏫 Introduction

Have you ever wondered how to measure the steepness of a hill at a single point? Or how fast something is moving at a specific second? 🎢 These questions are answered by tangents and gradients of curves.

In this lesson, you’ll learn how to estimate the slope (gradient) of a curve at a point by drawing a tangent line — a skill that bridges algebra and real-world motion.

By the end of this lesson, you’ll be able to:

• Understand what a tangent is and how it relates to a curve
• Estimate the gradient of a curve at a point using a tangent
• Apply this concept to solve practical problems

🧠 Key Concepts and Explanations

📌 What Is a Tangent?

A tangent to a curve at a point is a straight line that just touches the curve at that point, going in the same direction. It does not cut through the curve.

📈 What Is a Gradient?

The gradient (slope) of a straight line is:

On a curve, the gradient at a point is the slope of the tangent at that point.

🧭 Estimating Gradient on a Curve

1. Draw the graph of the function.
2. At the required point, draw a straight line that touches the curve and follows its direction — the tangent.
3. Pick two clear points on the tangent line.
4. Use the gradient formula to find the slope.

🧮 Sample Problem Walkthroughs

1. Estimate the gradient of the curve y = x² at x = 2.
   • Sketch or plot the curve y = x²
   • Draw a tangent at the point (2, 4)
   • Pick two points on the tangent (e.g., (1, 1) and (3, 9))
   • Calculate:
   • Estimated gradient = 4
2. Find the gradient of y = x² – 4x + 3 at x = 1.
   • Find the point: x = 1 → y = 1 – 4 + 3 = 0 → (1, 0)
   • Draw a smooth curve and a tangent at (1, 0)
   • Choose two points on the tangent (e.g., (0, -2) and (2, 2))
   • Calculate:
   • Estimated gradient = 2

✍🏽 Practice Exercises

🅰️ A. Identify and Sketch

1. Sketch the curve y = x² – 2x + 1 and draw a tangent at x = 2.
2. Sketch the curve y = -x² + 4x and draw a tangent at x = 1.

🅱️ B. Estimate Gradients

1. Using your sketch from A1, estimate the gradient at x = 2.
2. Using your sketch from A2, estimate the gradient at x = 1.

🆎 C. Word Connection

1. If y = x² represents distance (m) over time (s), what does the gradient at any point represent?

✅ Answers and Explanations

🅱️ B. Estimate Gradients (Example Approximations)

• 1. At x = 2 on y = x² – 2x + 1, point is (2, 1). Tangent might go through (1, 0) and (3, 4).
  
• 2. At x = 1 on y = -x² + 4x, point is (1, 3). Tangent through (0, 0) and (2, 4):
  

🆎 C. Concept Connection

• The gradient of a distance-time graph is speed. So at any point, it tells you the object’s speed at that exact moment.

📌 Recap

• ✅ A tangent touches a curve at one point without crossing it
• ✅ The gradient of the tangent gives the slope of the curve at that point
• ✅ You can estimate the gradient by drawing a tangent and applying the slope formula

💭 Reflection Prompt

Why do you think knowing the gradient at just one point on a curve is important in science or engineering?