Functions www mathsrevision com Nat 5 Functions Graphs

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

Starter Questions Nat 5 www. mathsrevision. com Q 1. Remove the brackets a (4 y – 3 x) Q 2. For the line y = -x + 5, find the gradient and where it cuts the y axis. Q 3. Find the highest common factor for p 2 q and pq 2. 16 -Oct-21 Created by Mr. Lafferty@mathsrevision. com

Functions www. mathsrevision. com Nat 5 Learning Intention 1. We are learning about functions and their associated graphs. 16 -Oct-21 Success Criteria 1. Understand the term function. 2. Know that the input is the xcoordinate and the output is the y-coordinate. 3. Recognise the graph of a linear and quadratic function. Created by Mr. Lafferty@www. mathsrevision. com

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

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

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

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

Function Notation Nat 5 www. mathsrevision. com 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 is the value of f at a

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

Function Notation Nat 5 Examples www. mathsrevision. com For the function g(x) = x 2 + x Calculate g(0) , g(3) and g(2 a) g(x) = x 2 + x g(0) = 02 + 0 = 0 g(3) = 32 + 3 = 12 g(2 a) = (2 a)2 +2 a = 4 a 2 + 2 a

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

f(x) =3 x + 1 Straight Line Functions y 10 9 x -1 0 1 2 y -2 1 4 7 10 8 7 6 5 4 3 2 1 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 -1 -2 -3 -4 -5 -6 -7 -8 -9 -10 16 -Oct-21 Created by Mr. Lafferty Maths Dept 8 9 10 x 3

Sketching Function Nat 5 www. mathsrevision. com 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.

y = x 2 - 3 Quadratic Functions y x 10 y 9 Demo 8 7 6 5 4 3 2 1 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 -1 -2 -3 -4 -5 -6 -7 -8 -9 -10 16 -Oct-21 Created by Mr. Lafferty Maths Dept 8 9 10 x -3 -1 6 -2 0 1 3 -3 -2 6

Function & Graphs www. mathsrevision. com Nat 5 Now try N 5 TJ Ex 12. 1 up to Q 9 Ch 12 (page 117) 16 -Oct-21 Created by Mr. Lafferty@www. mathsrevision. com

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

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

Function & Graphs www. mathsrevision. com Nat 5 Now try N 5 TJ Ex 12. 1 Q 10 onwards Ch 12 (page 117) 16 -Oct-21 Created by Mr. Lafferty@www. mathsrevision. com

Starter Questions www. mathsrevision. com Nat 5 16 -Oct-21 Created by Mr. Lafferty Maths Dept.

Graphs of linear and Quadratic functions www. mathsrevision. com Nat 5 Learning Intention 1. We are learning about linear and quadratic functions. Success Criteria 1. Understand linear and quadratic functions. 2. Be able to graph linear and quadratic equations. 16 -Oct-21 Created by Mr. Lafferty@www. mathsrevision. com

Graphs of linear and Quadratic functions Nat 5 It shows the link between the numbers in the input x ( or domain ) and output y ( or range ) 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 Output (Range) www. mathsrevision. com A graph gives a picture of a function c = 0 in this example ! Input (Domain) x

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

A function of the form f(x) = ax 2 + bx +c a ≠ 0 Graph Quadratic Function is calledof a quadratic function www. mathsrevision. com Nat 5 and its graph is a parabola with equation y = ax 2 + bx + c The parabola shown here is the graph of the function f defined by f(x) = x 2 + 2 x - 3 Its equation is y = x 2 + 2 x - 3 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)

Sketching Quadratic Functions Nat 5 www. mathsrevision. com Example : Sketch f(x) = x 2 { -3 ≤ x ≤ 3 } Make a table x y -3 -2 -1 0 1 2 3 9 4 1 0 1 4 9

What is the equation of symmetry ? y x x 10 9 Outcome 2 8 7 x=0 6 x x 5 4 3 2 x 1 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 -1 This function has one root. What is it ? 16 -Oct-21 7 8 9 10 x -2 -3 (0, 0) -4 -5 -6 -7 -8 x=0 6 -9 -10 What is the minimum turning point ? Created by Mr. Lafferty Maths Dept

Sketching Nat 5 Quadratic Functions www. mathsrevision. com Example : Sketch f(x) = 4 x – x 2 { -1 ≤ x ≤ 5 } Make a table x -1 y -5 0 1 2 3 0 3 4 5 0 -5

What is the equation of symmetry ? y 10 9 Outcome 2 8 7 x=2 6 x 5 4 3 x x 2 x 1 -10 -9 -8 -7 -6 -5 -4 -3 -2 0 -1 1 2 3 x 4 5 -1 6 7 8 9 10 x -2 -3 What are the roots of the function ? x -4 -5 -8 16 -Oct-21 (2, 4) -6 -7 x = 0 and 4 x -9 -10 What is the maximum turning point ? Created by Mr. Lafferty Maths Dept

Function & Graphs www. mathsrevision. com Nat 5 Now try N 5 TJ Ex 12. 2 Ch 12 (page 120) 16 -Oct-21 Created by Mr. Lafferty@www. mathsrevision. com