Graphs and Functions for GCSE

Introduction

Graphs and functions are integral to GCSE Maths, testing your ability to plot, interpret, and solve equations graphically. This article will cover:

1. Plotting and interpreting linear graphs.
2. Understanding quadratic and exponential graphs.
3. Solving equations graphically.

1. Plotting and Interpreting Linear Graphs

   Linear graphs have the equation \( y = mx + c \), where:

   • \( m \): Gradient (slope).
   • \( c \): Y-intercept (where the graph crosses the y-axis).

   Example: Plot \( y = 2x + 1 \).

   1. Create a table of values:
      • For \( x = 0 \), \( y = 1 \).
      • For \( x = 1 \), \( y = 3 \).
      • For \( x = -1 \), \( y = -1 \).
   2. Plot the points: \( (0,1), (1,3), (-1,-1) \).

2. Understanding Quadratic Graphs

   Quadratics have the form \( y = ax^2 + bx + c \).

   Key Features:

   1. Parabolic Shape: U-shaped or inverted.
   2. Vertex: Highest or lowest point.
   3. Axis of Symmetry: Vertical line through the vertex.

   Example: Plot \( y = x^2 – 4 \).

   1. Create a table:
      • \( x = -2, y = 0 \).
      • \( x = 0, y = -4 \).
      • \( x = 2, y = 0 \).
   2. Plot points and draw the parabola.

3. Solving Equations Graphically

   Graphs can help solve equations by finding where they intersect.

   Example: Solve \( x^2 – 4 = 0 \).

   1. Plot \( y = x^2 – 4 \).
   2. Find points where \( y = 0 \).
   3. Solutions: \( x = -2, x = 2 \).

Practice Question

Question: Plot \( y = -x + 3 \). Identify the gradient and y-intercept.

Solution:

• Gradient (\( m \)) = -1.
• Y-intercept (\( c \)) = 3.
• Plot the graph using points: \( (0,3), (1,2), (-1,4) \).

Conclusion

Mastering graphs and functions equips you to solve equations visually and interpret real-world scenarios. Practise regularly to ace GCSE graph-related questions.