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 functions from the general form to either the factorised form or the completed square form. ➢ The factorised form a(x – p)(x – q), where p and q are the x-intercepts ➢ The completed square form a(x – h) + k, where (h, k) are the coordinates of the turning point. Discriminant (b2 – 4ac) Nature of Roots of ax2 + bx + c = 0 Numbers of x-intercept Graph of y = ax2 + bx + c > 0 2 Real and Distinct Roots 2 = 0 2 Real and Equal (Repeated) Roots 1 < 0 No Real Roots 0 Turning Point Turning Point

5 Chapter 1 Quadratic Functions Secondary 3 Additional Mathematics Example 5 1. (i) Find the coordinates of the turning point and the y-intercept on the graph of y = –2x2 + 4x + 4. (ii) Find the exact values of x for which y = 0. Practice 5 1. (i) Find the coordinates of the turning point and the y-intercept on the graph of y = 4x2 – 12x + 9. (ii) Solve the equation y = 0. Example 6 1. (i) Find the x-intercepts and the coordinates of the turning point on the graph of y = 2x2 – 6x – 8. (ii) Hence explain why 2x2 – 6x – 8 = 0 has two distinct real roots.

6 Chapter 1 Quadratic Functions Secondary 3 Additional Mathematics Practice 6 1. (i) Find the x-intercepts and the coordinates of the turning point on the graph of y = –2x2 + 2x + 4. (ii) Hence explain why –2x2 + 2x + 4= 0 has two distinct real roots. Example 7 1. Use the discriminant to determine the number of x-intercepts for the graph of each given quadratic function. (a) y = x2 + 6x + 9 (b) y = (2 – x)2 + 1 Practice 7 1. Use the discriminant to determine the number of x-intercepts for the graph of each given quadratic function. (a) y = x2 + 2x – 6 (b) y = (1 – x)2 + 2

7 Chapter 1 Quadratic Functions Secondary 3 Additional Mathematics Example 8 1. Find the non-zero values of p for which the graph y = px2 – 2x + p touches the x-axis. Practice 8 1. Find the non-zero values of p for which the graph y = 3x2 + 12 – px meets only the x-axis at one point only. Example 9 1. Explain why the graph of y = x2 + (1 – p)x – p intersects the x-axis. Practice 9 1. Explain why the graph of y = x2 – 2x + 2 – p, where p > 1, intersects the x-axis at two points.

8 Chapter 1 Quadratic Functions Secondary 3 Additional Mathematics Example 10 1. (a) Show that the graph of y = x2 – 2kx + (k2 + 1) is entirely above the x-axis. (b) Show that –2x2 + 4kx – 2k2 – 1 is negative for all real values of x. Practice 10 1. (a) Show that the graph of y = –x2 + 2kx – (k2 + 1) is entirely above the x-axis. (b) Show that 1 4 x2 + kx + k2 + 1 is always positive.

9 Chapter 1 Quadratic Functions Secondary 3 Additional Mathematics 1.3 Applications of Quadratic Functions Example 11 1. Projectile Motion. A ball was launched from a slingshot. Its height, h m, above the ground is given by h = –2x2 + 8x + 1, where x m is the horizontal distance from the slingshot. (i) Find the height of the ball above the ground when it just left the slingshot. (ii) Find the greatest height of the ball after it was launched from the slingshot. (iii) If a toy is 3 m horizontally from the slingshot and 7 m above the ground, justify if the ball will hit the toy directly. Practice 11 End of Chapter 1 1.