Functions

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

2. 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 p2q and pq2. Created by Mr. Lafferty@mathsrevision.com

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

4. What are Functions ? Functions describe how one quantity relates to another Car Parts Cars Assembly line

5. What are Functions ? Functions describe how one quantity relates to another Dirty Clean Washing Machine y = f(x) x y Function Input Output f(x)

6. Finding the Function Examples Find the output or input values for the functions below : 6 7 8 36 49 64 4 12 f: 0 f: 1 f:2 -1 3 7 5 15 6 18 f(x) = x2 f(x) = 4x - 1 f(x) = 3x

7. 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. Function Notation The standard way to represent a function is by a formula. 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) = 1 + 4 = 5 5 is the value of f at 1 f(a) = a + 4 a + 4 is the value of f at a

9. Function Notation Examples For the function h(x) = 10 – x2. Calculate h(1) , h(-3) and h(5) h(x) = 10 – x2 h(1) = 10 – 12 = 9 h(-3) = 10 – (-3)2 = 10 – 9 = 1 h(5) = 10 – 52 = 10 – 25 = -15

10. Function Notation Examples For the function g(x) = x2 + x Calculate g(0) , g(3) and g(2a) g(x) = x2 + x  g(0) = 02 + 0 = 0 g(3) = 32 + 3 = 12 g(2a) = (2a)2 +2a = 4a2 + 2a

11. Sketching Function 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.

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

13. Sketching Function Example Consider the function f(x) = x2 - 3 and the set of x-values { -3, -1 , 0 , 1 , 3 } Find the value of f(-3) ....f(3) and plot them.

14. y 10 9 8 7 6 5 4 3 2 1 x 0 -10 1 2 3 4 5 6 7 9 10 -9 -8 -6 -4 -3 -2 8 -7 -5 -1 -1 -2 -3 -4 -5 -6 -7 -8 -9 -10 y = x2 - 3 Quadratic Functions x y -3 -1 0 1 3 6 -2 -3 -2 6 Demo Created by Mr. Lafferty Maths Dept

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

16. Finding the Function Example : Consider the function f(x) = x - 4 (a) Find an expression for f(3a) 3a ( ) - 4 3a - 4 Example : Consider the function f(x) = 3x2 + 2 (b) Find an expression for f(2p) 2p 3( )2 + 2 3(4p2) + 2 12p2 + 2

17. Finding the Function Remember 4 x 4 =16 Also (-4)x(-4) = 16 Example : Consider the function f(x) = x2 + 6 (a) Write down the value of f(k) k2 + 6 (b) If f(k) = 22 , set up an equation and solve for k. k2 + 6 = 22 k2 = 16 k = √16 k = 4 and - 4

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

19. Starter Questions Created by Mr. Lafferty Maths Dept.

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

21. Graphs of linear and Quadratic functions y 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 ) x A function of the form f(x) = mx + c is a linear function. c = 0 in this example ! Output (Range) Its graph is a straight line with equation y = mx + c Input (Domain)

22. Roots f(x) = x2 + 4x + 3 f(-2) =(-2)2 + 4x(-2) + 3 = -1 (0, ) a > 0 Mini. Point x= Line of Symmetry half way between roots Evaluating Graphs Quadratic Functions y = ax2 + bx + c c c Max. Point (0, ) a < 0 x= Line of Symmetry half way between roots

23. A function of the form f(x) = ax2 + bx +c a ≠ 0 is called a quadratic function and its graph is a parabola with equation y = ax2 + bx + c Graph of Quadratic Function The parabola shown here is the graph of the function f defined by f(x) = x2 + 2x - 3 Its equation is y = x2 + 2x - 3 • From the graph we can see • f(x) = 0 the roots are at • x = -3 and x = 1 • The axis of symmetry is half way between roots The linex = -1 • Minimum Turning Point of f(x) is half way between roots  (-1,-4)

24. Sketching Quadratic Functions Example : Sketch f(x) = x2 { -3 ≤ x ≤ 3 } Make a table 9 4 1 0 1 4 9

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

26. Sketching Quadratic Functions Example : Sketch f(x) = 4x – x2 { -1 ≤ x ≤ 5 } Make a table -5 -5 0 3 4 3 0

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

28. Function & Graphs Now try N5 TJ Ex 12.2 Ch12 (page120) Created by Mr. Lafferty@www.mathsrevision.com