📈 Functions & Graphs
You need to sketch and interpret linear, quadratic, power, and exponential graphs. Key skills include finding gradients, y-intercepts, turning points, and using tangent lines to estimate gradients at a point. Graph questions appear in both papers and can carry heavy marks in Paper 2.
Understanding graphs is about connecting algebra to geometry. Every equation has a corresponding graph, and every graph tells a story about the equation. For example, the roots of an equation are the x-intercepts of the graph, the gradient tells you how steep the line is, and the turning point of a parabola tells you the maximum or minimum value. Mastering this connection is essential for scoring well.
Linear Graphs: y = mx + c
A linear equation y = mx + c produces a straight-line graph. The constant m is the gradient (slope) of the line — it tells you how steep the line is and whether it slopes upward or downward. The constant c is the y-intercept — where the line crosses the y-axis. Two key relationships: parallel lines have the same gradient, and perpendicular lines have gradients that multiply to give $-1$.
📈Linear Functions
Straight-line graphs. Key concepts: gradient, y-intercept, equation of a straight line.
y = mx + cm = \frac{y_2 - y_1}{x_2 - x_1}y - y_1 = m(x - x_1)m_1 = m_2m_1 \times m_2 = -10 = mx + c \quad \Rightarrow \quad x = -\frac{c}{m}📝Worked Example 1: Finding the Equation of a Line
Find the equation of the line passing through A(1, 3) and B(4, 9).
y = 2x + 1 showing gradient, y-intercept, and x-intercept
📝Worked Example 2: Perpendicular Lines
Line L1 has equation y = 3x - 1. Find the equation of line L2 perpendicular to L1 passing through (6, 2).
L2: y = -x/3 + 4 showing gradient, y-intercept, and x-intercept
📝Worked Example 3: Finding x-intercept and y-intercept
Find the x-intercept and y-intercept of the line 3x - 2y = 12.
y = 3x/2 - 6 showing gradient, y-intercept, and x-intercept
What the full chapter covers
Linear Graphs: y = mx + c
Quadratic Graphs: y = ax^2 + bx + c
Exponential & Reciprocal Graphs
Exponential Graphs
Reciprocal Graph: y = 3/x
Gradient of a Curve at a Point
Speed-Time Graphs: Area and Gradient
Graph Transformations
Graph Sketching Checklist
Important: Sketch vs Draw vs Plot
Graph Interpretation Summary
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