Math Hl Type 1 Shadows
Higher Level Mathematics
Internal Assessment Type I
Shadow Functions
Contents
Introduction: Functions/Polynomials 3
Part A: Quadratic Polynomials 4
Part B: Cubic Polynomials 12
Introduction:
In mathematics, function is defined as a relationship, or more of a correspondence between the set of input values and the set of output values. Also, a rule is involved, or as it may be referred to, a ‘set of ordered pairs’ that assigns a unique output for each of the input. The output correspondence is usually defined as f and the output is x. The correspondence is denoted as f(x). All functions are mainly defined by two factors, as was mentioned before, set of inputs - which are called arguments; and outputs - which are oftenly called values. The set of all arguments is called domain; and the set of all the values is called range.
The graph on the left is just an example of a simple function f(x)=3x+2. As you can see it is a straight line.
Any function has an ability to be described through its relations to other functions, for example as an inverse function, or as a solution to a differential equation. Also, as we will see further, functions can be quadratic.
Quadratic function graph is simply a parabola. The equation of that function doesn’t contain any powers that are higher than 2. Basically, quadratic function is a polynomial of degree of 2.
There are different types of polynomials, but the most common ones and with what we are going to be working further, quadratic and cubic type of functions with polynomial equations of e.g. x2-2x+12 or x3-2x2-3x+13 etc.
Also, there is another type of functions, The Shadow Functions, which aren’t commonly used in everyday life, though still do deserve attention. Generally, Shadow Function is a simple reflection of an original function but via shadow generating function that shares the same vertex as the original and the shadow functions.Those are going to be investigated further in this
Internal Assessment Type I
Shadow Functions
Contents
Introduction: Functions/Polynomials 3
Part A: Quadratic Polynomials 4
Part B: Cubic Polynomials 12
Introduction:
In mathematics, function is defined as a relationship, or more of a correspondence between the set of input values and the set of output values. Also, a rule is involved, or as it may be referred to, a ‘set of ordered pairs’ that assigns a unique output for each of the input. The output correspondence is usually defined as f and the output is x. The correspondence is denoted as f(x). All functions are mainly defined by two factors, as was mentioned before, set of inputs - which are called arguments; and outputs - which are oftenly called values. The set of all arguments is called domain; and the set of all the values is called range.
The graph on the left is just an example of a simple function f(x)=3x+2. As you can see it is a straight line.
Any function has an ability to be described through its relations to other functions, for example as an inverse function, or as a solution to a differential equation. Also, as we will see further, functions can be quadratic.
Quadratic function graph is simply a parabola. The equation of that function doesn’t contain any powers that are higher than 2. Basically, quadratic function is a polynomial of degree of 2.
There are different types of polynomials, but the most common ones and with what we are going to be working further, quadratic and cubic type of functions with polynomial equations of e.g. x2-2x+12 or x3-2x2-3x+13 etc.
Also, there is another type of functions, The Shadow Functions, which aren’t commonly used in everyday life, though still do deserve attention. Generally, Shadow Function is a simple reflection of an original function but via shadow generating function that shares the same vertex as the original and the shadow functions.Those are going to be investigated further in this