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REVISION NOTES

IGCSE Edexcel Further Pure Mathematics

1.4 Graphs

1.4.1 Graphs of polynomials and rational functions with linear denominators

edexcel_igcse_further pure maths_fpm_topic 04_graphs_001_graphs of polynomial functions.png

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.

edexcel_igcse_further pure maths_fpm_topic 04_graphs_005_intersection of two linear graphs.png
edexcel_igcse_further pure maths_fpm_topic 04_graphs_009_gradient of a linear graph.png

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.

edexcel_igcse_further pure maths_fpm_topic 04_graphs_006_quadratic graphs.png

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.

edexcel_igcse_further pure maths_fpm_topic 04_graphs_002_characteristics of functions.png
edexcel_igcse_further pure maths_fpm_topic 04_graphs_003_characteristics of functions simplified.png

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.

edexcel_igcse_further pure maths_fpm_topic 04_graphs_004_graphs of reciprocal functions.png

Exponential and Logarithmic Function

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

edexcel_igcse_further pure maths_fpm_topic 04_graphs_008_exponentials and logarithm graphs.png

Trigonometric Function

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

edexcel_igcse_further pure maths_fpm_topic 04_graphs_007_sin cos tan graphs.png

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).

Back
Next
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

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