1. www.mathsrevision.com Nat 5 www.mathsrevision.com Functions Functions & Graphs Composite Functions Exam Type Questions See Quadratic Functions section The Quadratic Function

2. www.mathsrevision.com Nat 5 18-Aug-15Created by Mr. Lafferty@mathsrevision.com Starter Questions Q1.Remove the brackets a (4y – 3x) Q2.For the line y = -x + 5, find the gradient and where it cuts the y axis. Q3.Find the highest common factor for p 2 q and pq 2.

3. 18-Aug-15 Created by Mr. Lafferty@www.mathsrevision.com Learning Intention Success Criteria 1.Understand the term function. 1.We are learning about functions and their associated graphs. 2.Know that the input is the x- coordinate and the output is the y-coordinate. www.mathsrevision.com Functions Nat 5 3. Recognise the graph of a linear and quadratic function.

4. www.mathsrevision.com Nat 5 What are Functions ? Functions describe how one quantity relates to another Car Parts Assembly line Cars

5. www.mathsrevision.com Nat 5 What are Functions ? Functions describe how one quantity relates to another Dirty Washing Machine Clean OutputInput yx Function f(x) y = f(x)

6. www.mathsrevision.com Nat 5 Finding the Function Find the output or input values for the functions below : 6 7 8 36 49 64 f(x) = x 2 f: 0 f: 1 f:2 -1 3 7 f(x) = 4x - 1 4 12 f(x) = 3x 5 15 6 18 Examples

7. www.mathsrevision.com Nat 5 Defining a Functions A function can be thought of as the relationship between Set A (INPUT - the x-coordinate) and SET B the y-coordinate (Output).

8. www.mathsrevision.com Nat 5 The standard way to represent a function is by a formula. Function Notation Example f(x) = x + 4 We read this as “f of x equals x + 4” or “the function of x is x + 4 f(1) =5 is the value of f at 1 f(a) =a + 4 is the value of f at a 1 + 4 =5 a + 4

9. www.mathsrevision.com Nat 5 For the function h(x) = 10 – x 2. Calculate h(1), h(-3) and h(5) h(1) = Examples h(-3) = h(5) = h(x) = 10 – x 2  Function Notation 10 – 1 2 = 9 10 – (-3) 2 = 10 – 9 = 1 10 – 5 2 =10 – 25 = -15

10. www.mathsrevision.com Nat 5 For the function g(x) = x 2 + x Calculate g(0), g(3) and g(2a) g(0) = Examples g(3) = g(2a) = g(x) = x 2 + x  Function Notation 0 2 + 0 =0 3 2 + 3 = 12 (2a) 2 +2a =4a 2 + 2a

11. www.mathsrevision.com Nat 5 We will be using a formula to represent a function f(x)h(x)g(x) Example Consider the function f(x) = 3x + 1 and the set of x-values { -1, 0, 1, 2,3 } Find the value of f(-1)....f(3) and plot them. Sketching Function

12. 18-Aug-15Created by Mr. Lafferty Maths Dept 0 1234567 8 910 -9-8 -7 -6 -5 -4-3-2 -10 x y 1 2 3 4 5 6 7 8 9 10 -2 -3 -4 -5 -6 -7 -8 -9 -10 f(x) =3x + 1 xyxy 0 1 1 4 2 7 Straight Line Functions 3 10 -2

13. www.mathsrevision.com Nat 5 Example Consider the function f(x) = x 2 - 3 and the set of x-values { -3, -1, 0, 1, 3 } Find the value of f(-3)....f(3) and plot them. Sketching Function

14. 18-Aug-15Created by Mr. Lafferty Maths Dept 0 1234567 8 910 -9-8 -7 -6 -5 -4-3-2 -10 x y 1 2 3 4 5 6 7 8 9 10 -2 -3 -4 -5 -6 -7 -8 -9 -10 y = x 2 - 3 xyxy -2 0 -3 1 -2 Quadratic Functions 3 6 -3 6 Demo

15. www.mathsrevision.com Nat 5 18-Aug-15Created by Mr. Lafferty@www.mathsrevision.com Now try N5 TJ Ex 12.1 up to Q9 Ch12 (page117) Function & Graphs

16. www.mathsrevision.com Nat 5 Finding the Function Example :Consider the function f(x) = x - 4 (a) Find an expression for f(3a) ( ) - 4 3a 3a - 4 (b) Find an expression for f(2p) Example :Consider the function f(x) = 3x 2 + 2 3( ) 2 + 2 2p 3(4p 2 ) + 2 12p 2 + 2

17. www.mathsrevision.com Nat 5 Finding the Function Example :Consider the function f(x) = x 2 + 6 (b) If f(k) = 22, set up an equation and solve for k. (a) Write down the value of f(k) k 2 + 6 = 22 k 2 = 16 k = √16 k = 4 and - 4 k 2 + 6 Remember 4 x 4 =16 Also (-4) x (-4) = 16

18. www.mathsrevision.com Nat 5 18-Aug-15Created by Mr. Lafferty@www.mathsrevision.com Now try N5 TJ Ex 12.1 Q10 onwards Ch12 (page117) Function & Graphs

19. www.mathsrevision.com Nat 5 18-Aug-15Created by Mr. Lafferty Maths Dept. Starter Questions

20. 18-Aug-15 Created by Mr. Lafferty@www.mathsrevision.com Learning Intention Success Criteria 1.Understand linear and quadratic functions. 1.We are learning about linear and quadratic functions. 2.Be able to graph linear and quadratic equations. www.mathsrevision.com Nat 5 Graphs of linear and Quadratic functions

21. www.mathsrevision.com Nat 5 A graph gives a picture of a function It shows the link between the numbers in the input x ( or domain ) and output y ( or range ) Graphs of linear and Quadratic functions A function of the form f(x) = mx + c is a linear function. Its graph is a straight line with equation y = mx + c y x Input (Domain) O u t p u t ( R a n g e ) c = 0 in this example !

22. Quadratic Functions y = ax 2 + bx + c Graphs Evaluating Roots Mini. Point (0, ) (0, ) Max. Point Line of Symmetry half way between roots Line of Symmetry half way between roots a > 0 a < 0 f(x) = x 2 + 4x + 3 f(-2) =(-2) 2 + 4 x (-2) + 3 = -1 x = x = cc

23. www.mathsrevision.com Nat 5 Its equation is y = x 2 + 2x - 3 The parabola shown here is the graph of the function f defined by f(x) = x 2 + 2x - 3 Graph of Quadratic Function From the graph we can see (i)f(x) = 0 the roots are at x = -3 and x = 1 (i)The axis of symmetry is half way between roots The line x = -1 (ii)Minimum Turning Point of f(x) is half way between roots  (-1,-4) A function of the form f(x) = ax 2 + bx +ca ≠ 0 is called a quadratic function and its graph is a parabola with equation y = ax 2 + bx + c

24. www.mathsrevision.com Nat 5 Sketching Quadratic Functions Example : Sketch f(x) = x 2 { -3 ≤ x ≤ 3 } Make a table x-3-20123 y 9410149

25. Outcome 2 0 1234567 8 910 -9-8 -7 -6 -5 -4-3-2 -10 x y 1 2 3 4 5 6 7 8 9 10 -2 -3 -4 -5 -6 -7 -8 -9 -10 x 18-Aug-15Created by Mr. Lafferty Maths Dept xx xx What is the equation of symmetry ? x = 0 What is the minimum turning point ? (0,0) This function has one root. What is it ? x = 0

26. www.mathsrevision.com Nat 5 Sketching Quadratic Functions Example : Sketch f(x) = 4x – x 2 { -1 ≤ x ≤ 5 } Make a table x012345 y 03430 -5 -5

27. Outcome 2 0 1234567 8 910 -9-8 -7 -6 -5 -4-3-2 -10 x y 1 2 3 4 5 6 7 8 9 10 -2 -3 -4 -5 -6 -7 -8 -9 -10 x 18-Aug-15Created by Mr. Lafferty Maths Dept x x x x What is the equation of symmetry ? x = 2 What is the maximum turning point ? (2,4) x x What are the roots of the function ? x = 0 and 4

28. www.mathsrevision.com Nat 5 18-Aug-15Created by Mr. Lafferty@www.mathsrevision.com Now try N5 TJ Ex 12.2 Ch12 (page120) Function & Graphs