Basic Introduction to AS Mathematics

AS Mathematics is part of A-Level Mathematics. At AS level, two papers are assessed:

• International AS Unit P1 (Pure Maths)
• International AS Unit PSM1 (Pure Maths, Statistics and Mechanics)

You can view all the detailed specifications here.

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Revision Resources

Chapter 2: Quadratic

Quadratic PDF Notes

Chapter 4: Functions and Graphs

Geometrical Interpretation of Algebraic Solutions

In mathematics, algebraic solutions to equations can be visualized geometrically using graphs. The roots of an equation correspond to the points where the graph of the function intersects the x-axis. For example, solving \(f(x) = 0\) geometrically means identifying the x-coordinates of these intersection points. This approach provides an intuitive understanding of solutions and their behavior.

Using Intersection Points of Graphs to Solve Equations

The intersection points of two functions’ graphs, \(y = f(x)\) and \(y = g(x)\), represent the solutions to the equation \(f(x) = g(x)\). At these points, both functions share the same x and y values. Graphing both functions allows for a visual method of identifying solutions.

How to Sketch a Quadratic Function

Quadratic functions take the form \( y = ax^2 + bx + c \). Follow these steps:

1. Identify the Shape
   • If \( a > 0 \), parabola opens upwards
   • If \( a < 0 \), parabola opens downwards
2. Find the Vertex
   • X-coordinate: \( x = -\frac{b}{2a} \)
   • Substitute to find y-coordinate
3. Determine the Axis of Symmetry
   • Vertical line \( x = -\frac{b}{2a} \)
4. Find the Y-Intercept
   • Set \( x = 0 \) → \( y = c \)
5. Find X-Intercepts
   • Solve \( ax^2 + bx + c = 0 \) using:
     • Factoring
     • Completing the square
     • Quadratic formula
6. Plot and Sketch
   • Combine key points and draw smooth curve

How to Sketch a Cubic Function

Cubic functions take the form \( y = ax^3 + bx^2 + cx + d \). Procedure:

1. End Behavior
   • \( a > 0 \): Rises right, falls left
   • \( a < 0 \): Falls right, rises left
2. Turning Points Analysis
   • First derivative: \( y’ = 3ax^2 + 2bx + c \)
   • Solve \( y’ = 0 \) for critical points

3. Δ (\( b^2 - 3ac \)) Interpretation

   Condition	Graph Characteristics
   \( Δ > 0 \)	Two turning points (S-shaped)
   \( Δ = 0 \)	One flattened turning point
   \( Δ < 0 \)	Monotonic (no turning points)

4. Key Points Identification
   • Y-intercept: \( x=0 \) → \( y=d \)
   • X-intercepts: Solve \( ax^3 + bx^2 + cx + d = 0 \)
   • Turning points: From derivative analysis
5. Graph Construction
   • Plot intercepts and critical points
   • Follow end behavior pattern
   • Draw smooth connecting curve

Credit to Lynn for all the PDFs