S4 Credit Functions Illustrating a Function Standard Notation for a Function f(x) Graphs of linear and Quadratic Functions Exponential Function Summary of Graphs and Functions Mathematical Modelling Sketching Quadratic Functions Reciprocal Function

S4 Credit 23-Nov-16Created by Mr. Starter Questions Q1.Remove the brackets (a)a (4y – 3x) =(b)(x + 5)(x - 5) = 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.

23-Nov-16 Created by Mr. Learning Intention Success Criteria 1.Understand the term function. 1.To explain what a function is in terms of a diagram and formula. 2.Apply knowledge to find functions given a diagram. Functions S4 Credit

S4 Credit What are Functions ? Functions describe how one quantity relates to another Car Parts Assembly line Cars

S4 Credit What are Functions ? Functions describe how one quantity relates to another Dirty Washing Machine Clean OutputInput yx Function f(x) y = f(x)

S4 Credit If the first set is A and the second B then we often write f: A  B Defining a Functions Defn: A function is a relationship between two sets in which each member of the first set is connected to exactly one member in the second set. The members of set A are usually referred to as the domain of the function (basically the starting values or even x-values) while the corresponding values come from set B and are called the range of the function (these are like y-values).

S4 Credit Functions can be illustrated in a number of ways: 1) by a formula. 2) by arrow diagram. Example Suppose that f: A  B is defined by f(x) = x 2 + 3x where A = { -3, -2, -1, 0, 1}. FORMULA then f(-3) = 0,f(-2) = -2, f(-1) = -2, f(0) = 0, f(1) = 4 NB: B = {-2, 0, 4} = the range! Illustrating Functions

S4 Credit A B ARROW DIAGRAM f(-3) = 0 f(-2) = -2 f(-1) = -2 f(0) = 0 f(1) = 4 f(x) = x 2 + x Illustrating Functions

S4 Credit Finding the Function Find the output or input values for the functions below : f(x) = x 2 f: 0 f: 1 f: f(x) = 4x f(x) = 3x Examples

S4 Credit Finding the Function Find the function f(x) for from the diagrams f(x) f: 0 f: 1 f: f(x) f(x) = x + 5f(x) = x 2 f(x) = 2x Examples

S4 Credit 23-Nov-16Created by Mr. Now try MIA Ex 2.1 Ch10 (page195) Illustrating Functions

S4 Credit 23-Nov-16Created by Mr. Lafferty12 Starter Questions Q1. Q2.Find the ratio of cos 60 o Q x 70 Q4. Explain why the length a = 36m 30m 24m a

23-Nov-16 Created by Mr. Learning Intention Success Criteria 1.Understand function notation. 1.To explain the mathematical notation when dealing with functions. 2.Be able to work with function notation. Function Notation S4 Credit

S4 Credit 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) = = 55 is the value of f at 1 f(a) = a + 4a + 4 is the value of f at a

S4 Credit For the function h(x) = 10 – x 2. Calculate h(1), h(-3) and h(5) h(1) = 10 – 1 2 = 9 Examples h(-3) = 10 – (-3) 2 = 10 – 9 = 1 h(5) = 10 – 5 2 = 10 – 25 = -15 h(x) = 10 – x 2  Function Notation

S4 Credit For the function g(x) = x 2 + x Calculate g(0), h(3) and h(2a) h(0) = = 0 Examples h(3) = = 12 h(2a) = (2a) 2 +2a = 4a 2 + 2a g(x) = x 2 + x  Function Notation

S4 Credit 23-Nov-16Created by Mr. Now try MIA Ex 3.1 & 3.2 Ch10 (page197) Function Notation

S4 Credit 23-Nov-16Created by Mr. Lafferty Maths Dept. Starter Questions

23-Nov-16 Created by Mr. Learning Intention Success Criteria 1.Understand linear and quadratic functions. 1.To explain the linear and quadratic functions. 2.Be able to graph linear and quadratic equations. S4 Credit Graphs of linear and Quadratic functions

S4 Credit 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) = ax + b is a linear function. Its graph is a straight line with equation y = ax + b y x Input (Domain) O u t p u t ( R a n g e )

S4 Credit The parabola shown here is the graph of the function f defined by f(x) = x 2 – 1 for -4  x  4 It equation is y = x Graph of Quadratic Function From the graph we can see (i)Minimum value of f(x) is -1 at x = 0 (ii)Minimum turning point (TP) is (0,-1) (iii) f(x) = 0 at x = -1 and x = 1 (iv)The axis of symmetry is the line x = 0 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

Outcome 2 x x x y Nov-16Created by Mr. Lafferty Maths Dept y = 2x + 1 xyxy y = 2x - 5 xyxy 01 3 Draw the graph of the functions with equations below : y = 2 - x 2 xyxy -202 y = x 2 xyxy -220 xx xx

S4 Credit 23-Nov-16Created by Mr. Now try MIA Ex 4.1 & 4.2 Ch10 (page 201) Graphs of Linear and Quadratic Functions

S4 Credit 23-Nov-16Created by Mr. Lafferty24 Starter Questions Q1.Round to 2 significant figures Q2.Why is x 2 = 10 and not 12 Q3.Solve for x (a)52.567(b)626

23-Nov-16 Created by Mr. Learning Intention Success Criteria 1.Be able to sketch quadratic functions. 1.To show how to sketch quadratic functions. S4 Credit Sketching Quadratic Functions

S4 Credit Sketching Quadratic Functions We can use a 4 step process to sketch a quadratic function Example 1 : Sketch f(x) = 15 – 2x – x 2 Step 1 : Find where the function crosses the x – axis. i.e x - x 2 = 0 SAC Method (5 + x)(3 - x) = 0 x = - 5x = x - x 5 + x = 03 - x = 0 (- 5, 0)(3, 0)

S4 Credit Step 2 : Find equation of axis of symmetry. It is half way between points in step 1 Sketching Quadratic Functions Equation is x = -1 Step 3 : Find coordinates of Turning Point (TP) For x = -1f(-1) = 15 – 2 x (-1) – (-1) 2 = 16 Since (-5, 0) and (3,0) lie on the curve and 0 is less than 16 Turning point TP is a Maximum at (-1, 16) (-5 + 3) ÷ 2 = -1

S4 Credit 3 -5 Step 4 :Find where curve cuts y-axis. For x = 0f(0) = 15 – 2 x 0 – 0 2 = 15 (0,15) X Y Cuts x-axis at -5 and 3 Cuts y-axis at 15 Max TP (-1,16)(-1,16) 15 Sketching Quadratic Functions Now we can sketch the curve y = 15 – 2x – x 2

S4 Credit Sketching Quadratic Functions We can use a 4 step process to sketch a quadratic function Example 2 : Sketch f(x) = x 2 - 7x + 6 Step 1 : Find where the function crosses the x – axis. i.e. x 2 – 7x + 6 = 0 SAC Method (x - 6)(x - 1) = 0 x = 6x = 1 x x x - 6 = 0x - 1 = 0 (6, 0)(1, 0)

S4 Credit Step 2 : Find equation of axis of symmetry. It is half way between points in step 1 Sketching Quadratic Functions Equation is x = 3.5 Step 3 : Find coordinates of Turning Point (TP) For x = 3.5 f(3.5) = (3.5) 2 – 7 x (3.5) + 6 = Since (1, 0) and (6,0) lie on the curve and 0 is greater than 2.25 Turning point TP is a Minimum at (3.5, -2.25) (6 + 1) ÷ 2 =3.5

S4 Credit Cuts x - axis at 1 and Step 4 :Find where curve cuts y-axis. For x = 0f(0) = 0 2 – 7 x 0 = 6 = 6 (0,6) X Y Cuts y - axis at 6 Mini TP (3.5,-2.25)(3.5,-2.25) 6 Sketching Quadratic Functions Now we can sketch the curve y = x 2 – 7x + 6

S4 Credit 23-Nov-16Created by Mr. Now try MIA Ex 5.1 Ch10 (page 204) Sketching Quadratic Functions

S4 Credit 23-Nov-16Created by Mr. Lafferty Maths Dept. Starter Questions Q1. Explain why 15% of £80 is £12 Q3. Q2. Multiply out the brackets Q4.

23-Nov-16 Created by Mr. Learning Intention Success Criteria 1.Know the main points of the reciprocal function. 1.To show what the reciprocal function looks like. 2.Be able to sketch the reciprocal function. S4 Credit The Reciprocal Function

S4 Credit The function of the form is the simplest form of a reciprocal function. The Reciprocal Function The graph of the function is called a hyperbola and it divided into two branches. The equation of the graph is y is inversely proportional to x

S4 Credit The Reciprocal Function Note that x CANNOT take the value 0. The graph never touches the x or y axis. The axes are said to be asymptotes to the graph The graph has two lines of symmetry at 45 0 to the axes y x

x y y = 1/x xyxy Nov-16Created by Mr. Lafferty Maths Dept Draw the graph of the function with equations below : x x x y = 1/(-10) = y = 1/(-1) = - 1 y = 1 /(-0.1) = - 10 y = 1 / 0.1 = 10 y = 1 / 1 = 1 y = 1 / 10 = 0.1

x y y = 5/x xyxy Nov-16Created by Mr. Lafferty Maths Dept Draw the graph of the function with equations below : x x x y = 5/(-10) = y = 5/(-5) = - 1 y = 5/(-1) = - 5 y = 5 / 1 = 5 y = 5 / 5 = 1 y = 5 / 10 = 0.5

S4 Credit 23-Nov-16Created by Mr. Now try MIA Ex 6.1 Ch10 (page 206) Reciprocal Function

S4 Credit 23-Nov-16Created by Mr. Lafferty Maths Dept. Starter Questions Q1. True or false 2a (a – c + 4ab) =2a 2 -2ac + 8ab Q3. Q2. Find the missing angle Q4. 22 o

23-Nov-16 Created by Mr. Learning Intention Success Criteria 1.Know the main points of the exponential function. 1.To show what the exponential function looks like. 2.Be able to sketch the exponential function. S4 Credit Exponential Function

S4 Credit A function in the form f(x) = a x where a > 0, a ≠ 1 is called an exponential function to base a. Exponential Functions Consider f(x) = 2 x x f(x) 11 / 8 ¼ ½ Exponential (to the power of) Graphs

x y Nov-16Created by Mr. Lafferty Maths Dept Draw the graph of the function with equation below : x x x x = -3y = 1/8 x = -2y = 1/4 x = -1y = 1/2 x = 0y = 1 x = 1y = 2 x = 2y = 4 x y = 2 x x = 3y = 8

S4 Credit The graph is like y = 2 x (0,1) (1,2) Major Points (i) y = 2 x passes through the points (0,1) & (1,2) (ii) As x  ∞ y  ∞ however as x  ∞ y  0. (iii) The graph shows a GROWTH function. Graph

S4 Credit Remember We can calculate exponential (power) value on the calculator. Exponential Button on the Calculator Button looks like yxyx Examples Calculate the following 2 5 = 3 -2 = = 5y x 2 = -y x /

S4 Credit 23-Nov-16Created by Mr. Now try MIA Ex 7.1 Ch10 (page 208) Exponential Function

S4 Credit 23-Nov-16Created by Mr. Lafferty Maths Dept. Starter Questions 39 o

23-Nov-16 Created by Mr. Learning Intention Success Criteria 1.Know the main points of the various graphs in this chapter. 1.To summarise graphs covered in this chapter. 2.Be able to identify function and related graphs. S4 Credit Summary of Graphs & Functions

S4 Credit 23-Nov-16Created by Mr. Summary of Graphs & Functions y x x Y y x f(x) = ax + b g(x) = ax 2 + bx + c h(x) = a / x k(x) = a x Linear Quadratic Reciprocal Exponential Y x

S4 Credit 23-Nov-16Created by Mr. Now try MIA Ex 8.1 Ch10 (page 209) Summary of Graphs & Functions

S4 Credit 23-Nov-16Created by Mr Lafferty Maths Dept Starter Questions

23-Nov-16 Created by Mr. Learning Intention Success Criteria 1.Understand mathematical models using functions 1.To show how we can use functions to model real-life situations. 2.Solving problems using mathematical models. S4 Credit Mathematical Models

S4 Credit 23-Nov-16Created by Mr. Mathematical Models In real-life scientists look for connections between two or more quantities. (A) They collect data, using experiments, surveys etc... (B) They organise the data using tables and graphs. (C) They analyse the data, by matching it with graphs like the ones you have studied so far. (D) Use the results to predict for results

S4 Credit 23-Nov-16Created by Mr. Mathematical Models Collect data Organise data Analyse data Make Predictions TablesGraphs SurveyExperiments

S4 Credit McLaren are testing a new Formula 1 car. Data was collected and organised into the table below: Example Mathematical Models By plotting the data on a graph and analysing the result is there a connection between the variables time and distance ?

x y x x x x = 0y = 0 x = 0.5y = 2 x = 1.0y = 8 x = 1.5y = 18 x = 2.0y = 32 x = 2.5y = 50 x y = ax 2 Does it look like part of a graph we know ? To find a pick a point of the graph and sub into equation. (1,8) y = ax 2 8 = a x (1) 2 y = 8x 2 Pick another point to double check ! (2,32)32 = 8 x (2) 2 32 = 8 x 432 = 32 x Y y = ax 2

S4 Credit We can now use the equation to predict other values. Example Mathematical Models Use the equation to calculate the following : y = 8x 2 distance when time x = 10 time when distance y = 200 y = 8x 2 y = 8 x (10) 2 y = 8 x 10 y = 800 m y = 8x = 8x 2 x 2 = 25 x = 5 seconds

S4 Credit 23-Nov-16Created by Mr. Now try MIA Ex 9.1 Ch10 (page 210) Mathematical Models