S3E A Math WS Chapter 1

1 Chapter 1 Quadratic Functions Secondary 3 Additional Mathematics Chapter 1 Quadratic Functions Name: ______________________________ ( ) Class: 3_______ Date: ________________________ Section Topic Page 1.1 Maximum and Minimum Values 1 1.2 Graphical Representation of Quadratic Functions 4 1.3 Applications of Quadratic Functions 9 1.1 Maximum and Minimum Values We use the notation y = f(x) to mean that y is a _____________ of f(x). To find all possible real values of f(x) = ax2 + bx + c, we first use the method of completing the square to express the function in the form ______________, where ____ and ____ are constants. Activity One: Sketch the graph of y = mx + c and the graph of y = ax2 + bx + c. Example 1 1. Given f(x)² = 2x² – 4, find (i) its value when x = 0, (ii) all possible real values of f(x). y = mx + c I observe that the graph y = mx + c is a ______________________________________ y = ax2 + bx + c I observe that the graph y = ax2 + bx + c is a ______________________________________

2 Chapter 1 Quadratic Functions Secondary 3 Additional Mathematics Practice 1 1. Given f(x)² = 2x² + 4, find (i) f(–1) , (ii) all possible real values of f(x). Example 2 1. Express f(x) = x² + 6x – 4 in the form (x – h)² + k, where h and k are constants. Hence find (i) all possible real values of f(x), (ii) the minimum value of f(x), (iii) the value of x at which the minimum value of f(x) occurs. Practice 2 1. Given f(x) = x² + 3x – 6, find (i) all possible real values of f(x), (ii) the minimum value of f(x), (iii) the value of x at which the minimum value of f(x) occurs.

3 Chapter 1 Quadratic Functions Secondary 3 Additional Mathematics Example 3 1. Express each of the following functions in the form a(x – h)² + k, where a, h and k are constants, and a ≠ 0. Hence find the minimum or maximum value of f(x). Justify your answers. (a) f(x) = 3x² – 6x + 2 (b) f(x) = –2x² – 3x + 3 Practice 3 1. Express each of the following functions in completed square form. Hence find the maximum or minimum value of f(x). Justify your answers. (a) f(x) = 2x² – 3x – 1 (b) f(x) = –3x² + 4x – 2 Example 4 1. Show that x² – 2x + 6 ≥ 5 for all real values of x. Practice 4 1. Show that –x² – 4x – 10 is negative for all real values of x.

4 Chapter 1 Quadratic Functions Secondary 3 Additional Mathematics 1.2 Graphical Representation of Quadratic Functions The graph of a quadratic function changes direction at a point called the turning point or the vertex of the graph. The general form ax2 + bx + c is not very useful in sketching quadratic graphs, as it does not show features like intercepts and turning points. As such, we would convert quadratic