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International Baccalaureate Mathematics

Functions

Syllabus Content

The quadratic function f(x)=ax2+bx+c: its graph, y-intercept (0,c). Axis of symmetry. The form f(x)=a(x-p)(x-q), x-intercepts (p,0) and (q,0). The form f(x)=a(x-h)2+k, vertex (h,k)

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Furthermore

Official Guidance, clarification and syllabus links:

A quadratic graph is also called a parabola.

Link to: transformations (SL 2.11).

Candidates are expected to be able to change from one form to another.

Formula Booklet:

Axis of symmetry of the graph of a quadratic function

\( f(x) = ax^2 + bx + c \\ \implies \text{ axis of symmetry is } x = - \dfrac{b}{2a}\)

The quadratic function \( f(x) = ax^2 + bx + c \) is a fundamental concept in algebra. Its graph forms a parabola, a symmetric curve that can open upwards or downwards depending on the coefficient \( a \). The y-intercept of this function is the point where the graph crosses the y-axis, which occurs at \( (0, c) \). An important feature of the parabola is its axis of symmetry, a vertical line that divides the graph into two mirror images. This axis passes through the vertex of the parabola, a point representing the maximum or minimum value of the function.

$$\text{ The axis of symmetry is } x = - \dfrac{b}{2a} $$

In the factored form \( f(x) = a(x - p)(x - q) \), the function is expressed as a product of two linear terms, where \( p \) and \( q \) are the x-intercepts of the graph, the points where the graph crosses the x-axis. These intercepts are found at \( (p, 0) \) and \( (q, 0) \). In the vertex (completing the square) form \( f(x) = a(x - h)^2 + k \), the function is represented in a way that highlights its vertex, located at \( (h, k) \). This form is particularly useful for easily identifying the vertex and for graphing the parabola.

Key Formulae:

$$ \text{Standard Form: } f(x) = ax^2 + bx + c \\ \text{Factored Form: } f(x) = a(x - p)(x - q) \\ \text{Vertex Form: } f(x) = a(x - h)^2 + k $$

Example:

Consider the quadratic function \( f(x) = 2x^2 - 8x + 6 \). Its standard form is already given. To find its factored form, we need to factorise the quadratic equation:

$$ f(x) = 2(x - 1)(x - 3) $$

The x-intercepts are \( (1, 0) \) and \( (3, 0) \). To express this function in vertex form, we complete the square:

$$ f(x) = 2(x - 2)^2 - 2 $$

The vertex of this parabola is at \( (2, -2) \).

This video on Completing The Square is from Revision Village and is aimed at students taking the IB Maths AA Standard level course

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