Sketching Curves

2026 Syllabus Objectives

By the end of this topic, you should be able to:

  1. Recognise, sketch and interpret graphs of linear and quadratic functions
  2. Understand the symmetry and roots of graphs (turning points knowledge is also required for quadratics)
  3. Recognise, sketch and interpret graphs of linear, quadratic, cubic, reciprocal, and exponential functions
  4. Work with specific function forms and understand turning points, roots, symmetry, and asymptotes
  5. Find turning points of quadratic graphs by completing the square

Main Content

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What Does "Sketching a Curve" Mean?

When you sketch a curve, you are drawing a simplified version of a graph that shows the most important features. You do not need to plot every single point carefully. Instead, you focus on:

  • Where the graph crosses the axes
  • The overall shape of the curve
  • Any special points (like turning points or asymptotes)
  • The general direction the curve goes

A sketch is different from an accurate plot because you are showing the main characteristics rather than exact positions of every point.

1. Linear Graphs

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What is a Linear Graph?

A linear graph is a straight-line graph. The equation of a linear graph can be written in the form:

ax + by = c

where a, b, and c are numbers.

You might also see linear equations written as y = mx + c, which is just another way of writing the same thing.

Key Features of Linear Graphs

  • Shape: Always a straight line
  • Gradient (slope): This tells you how steep the line is. A positive gradient means the line goes upwards from left to right. A negative gradient means it goes downwards.
  • y-intercept: This is where the line crosses the y-axis (when x = 0)
  • x-intercept: This is where the line crosses the x-axis (when y = 0)

How to Sketch a Linear Graph

Step 1: Find the y-intercept by setting x = 0 and solving for y

Step 2: Find the x-intercept by setting y = 0 and solving for x

Step 3: Plot these two points on a coordinate grid

Step 4: Draw a straight line through both points, extending it in both directions

Example: Sketch the graph of 2x + y = 6

  • y-intercept: When x = 0, we get 0 + y = 6, so y = 6. Point: (0, 6)
  • x-intercept: When y = 0, we get 2x + 0 = 6, so x = 3. Point: (3, 0)
  • Draw a straight line through (0, 6) and (3, 0)

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