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  • IGCSE Edexcel
  • Further Pure Mathematics
  • 1. Further Pure Mathematics Notes

Further Pure Mathematics

  • IGCSE Edexcel
  • Revision Notes

1. Further Pure Mathematics Notes


1.1 Logarithmic Functions and Indices>1.2 The Quadratic Function>1.3 Identities and Inequalities>1.4 Graphs>1.5 Series>1.6 The Binomial Series>1.7 Scalar and Vector Quantities>1.8 Rectangular Cartesian Coordinates>1.9 Calculus>1.10 Trigonometry>

1.4 Graphs

1.4.1 Graphs of polynomials and rational functions with linear denominators

Linear Function

Step 1: Plot the y-intercept

Step 2: Use the gradient to find the second coordinate

Step 3: Sketch the linear function

Don’t forget to label the x-axis, x- intercept, y-intercept.

Quadratic Function

Step 1: Find the y-intercept (x = 0)

Step 2: Find the x-intercept (y = 0) by solving the cubic function [Topic 2]

Step 3: Sketch the quadratic function

Don’t forget to label the x-axis, x- intercept, y-intercept, and turning point.

Cubic Functions

Step 1: Find the y-intercept (x = 0)

Step 2: Find the x-intercept (y = 0) by solving the cubic function [Topic 3 Part 2]

Step 3: Find the turning point by completing the square or differentiation

Step 4: Sketch the cubic function

Don’t forget to label the x-axis, x- intercept, y-intercept, and turning point.

Reciprocal Function

Sketching Reciprocal Function

Step 1: Determine the location of the curve (Positive – Quadrant 1 and 3) (Negative – Quadrant 2 and 4)

Step 2: Draw the asymptotes with a dotted line

  • Horizontal Asymptote
    • If the power of numerator < power of denominator, horizontal asymptote is the x-axis (y = 0)
    • If the power of numerator = power of denominator, horizontal asymptote is y = numerator leading coefficient / denominator leading coefficient
    • If the power of numerator > power of denominator, there is no horizontal asymptote
  • Vertical Asymptote
    • The denominator of a fraction cannot be 0
    • Square root cannot be less than 0

Step 3: Sketch the reciprocal function

Don’t forget to label the axis, x-intercept, y-intercept, horizontal asymptote, and vertical asymptote.

Exponential and Logarithmic Function

Exponential and logarithmic graph is a reflection  with y = x.

Trigonometric Function

Sin and Cos graph is periodic with a period of 360o . Tan graph is periodic with a period of 180o .

Graph Transformation

f(x + a) is a horizontal translation of – a
f(x) + a is a vertical translation of + a
f(ax) is a horizontal stretch of a scale factor 1/a
af(x) is a vertical stretch of scale factor a

1.4.2 The solution of equations and transcendental functions by graphical methods

Type 1: 2 Linear Equations

Type 2: Quadratic and Linear Equation

Part 2: Remainder Theorem

If a polynomial f(x) is divided by (ax – b), the remainder is f(b/a).

Previous1.3 Identities and Inequalities
Next1.5 Series

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