1031 Mathematics for Life Sciences

Anchor_Mathematics, 17/9/2025

Tabla de Contenidos

  1. Basic Functions
    1. Transformations of an unknown function f(x)
    2. Reflections of an unknown function
    3. Line in gradient intercept form
    4. Parabola in factored form
    5. Parabola from completed square
    6. Cubic in translation form
    7. Cubic in factored form
    8. Absolute Value
    9. Hyperbola - Inverse Variation
    10. Circle centre (h,k), radius r
    11. Polynomial degree 4 - quartic in factor form
    12. Polynomial degree 5 - quintic in factor form
  2. Trigonometric Functions
    1. Sine Functions - Sinusoids
    2. Auxiliary Angle Formula for Sine
    3. Phase of Sine Functions - Sinusoids
  3. Exponential and Logarithmic Functions
    1. Exponential graph with variable Base
    2. Log graph with variable base
    3. Exponential graph base e
    4. Log Natural graph base e
  4. Calculus
    1. Differentiation from First Principles - Visualisation
    2. Gradient of an Exponential Function
    3. Gradient of Sine Function
    4. Gradient of Cosine Function
  5. Least Squares and Approximation Models
    1. Finding the 'Line of Best Fit'
  6. Three Dimensional Space
    1. Planes defined by Cartesian Equation
    2. Multiple Planes defined by Cartesian Equation

1.

Basic Functions

This chapter show the common transformations of functions, as well as the graphs of the basic functions and relationships

  • 1. Transformations of an unknown function f(x)
  • 2. Reflections of an unknown function
  • 3. Line in gradient intercept form
  • 4. Parabola in factored form
  • 5. Parabola from completed square
  • 6. Cubic in translation form
  • 7. Cubic in factored form
  • 8. Absolute Value
  • 9. Hyperbola - Inverse Variation
  • 10. Circle centre (h,k), radius r
  • 11. Polynomial degree 4 - quartic in factor form
  • 12. Polynomial degree 5 - quintic in factor form

1.1.

Transformations of an unknown function f(x)

Translations and scaling/reflections of an unknown function f(x)
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1.2.

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1.3.

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1.5.

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1.6.

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1.7.

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1.8.

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1.9.

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1.10.

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1.11.

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1.12.

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2.

Trigonometric Functions

This chapter shows the basic sine and cosine graphs as well as their transformations

  • 1. Sine Functions - Sinusoids
  • 2. Auxiliary Angle Formula for Sine
  • 3. Phase of Sine Functions - Sinusoids

2.1.

Sine Functions - Sinusoids

Translations and scaling/reflections of the sine and cosine functions
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2.2.

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2.3.

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3.

Exponential and Logarithmic Functions

This chapter shows graphs of the exponential and logarithmic functions of any base, and their transformations. There are separate graphs for [b]the[/b] exponential function (base [i]e[/i]) and the natural logarithmic function.

  • 1. Exponential graph with variable Base
  • 2. Log graph with variable base
  • 3. Exponential graph base e
  • 4. Log Natural graph base e

3.1.

Exponential graph with variable Base

Graph in the form y = k(a^(x+b))+c
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3.2.

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3.3.

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3.4.

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4.

Calculus

  • 1. Differentiation from First Principles - Visualisation
  • 2. Gradient of an Exponential Function
  • 3. Gradient of Sine Function
  • 4. Gradient of Cosine Function

4.1.

Differentiation from First Principles - Visualisation

Visualisation of gradient between two points becoming gradient of curve at a single point
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4.2.

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4.4.

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5.

Least Squares and Approximation Models

The development of techniques to model datasets that do not exactly match a straight line, exponential curve, or power curve

  • 1. Finding the 'Line of Best Fit'

5.1.

Finding the 'Line of Best Fit'

The linear regression model is based on the sum of the squares of the vertical residuals minimised
How does the gradient (m) and y-intercept (b) of your line affect the residuals?
The Least Squares process to find a 'Line of Best Fit' to model your data depends on minimizing the sum of the squares of the vertical residuals.[br][br]Adjust the gradient and y-intercept values to try and find the best fit, then compare your answer to the linear regression model.[br][br]The six points A-F are also moveable.
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6.

Three Dimensional Space

  • 1. Planes defined by Cartesian Equation
  • 2. Multiple Planes defined by Cartesian Equation

6.1.

Planes defined by Cartesian Equation

The equation of the plane is ax+by+cz=d
Cartesian Equation and axes intercepts
Change the values of the coefficients a, b, c and d.[br][br]What happens when any of a, b, c are zero? Two of them? Three of them?[br][br]What happens when d = 0?
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6.2.

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