Sketching Quadratic Functions Illustrating a Function Standard Notation for a Function f(x) Graphs of linear and Quadratic Functions Sketching Quadratic Functions Reciprocal Function Exponential Function Summary of Graphs and Functions Mathematical Modelling Investigation of Area / Perimeter Exam Type Questions www.mathsrevision.com

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 p2q and pq2. 14-Dec-17

Functions www.mathsrevision.com Learning Intention Success Criteria S4 Credit Learning Intention Success Criteria To explain what a function is in terms of a diagram and formula. Understand the term function. Apply knowledge to find functions given a diagram. www.mathsrevision.com 14-Dec-17

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

Functions describe how one quantity 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)

Defining a Function 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. If the first set is A and the second B then we often write f: A  B 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).

Illustrating Functions Functions can be illustrated in a number of ways: 1) by a formula. 2) by arrow diagram. Domain Example Suppose that f: A  B is defined by f(x) = x2 + 3x where A = { -3, -2, -1, 0, 1}. FORMULA then f(-3) = f(-2) = f(-1) = f(0) = f(1) = -2 4 -2 Range B = {-2, 0, 4}

Illustrating Functions ARROW DIAGRAM A B f(x) = x2 + 3x f(-3) = 0 f(-2) = -2 f(-1) = -2 f(0) = 0 f(1) = 4 -3 -2 -2 -1 -2 1 4

Find the output or input values for the functions below : 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

Find the function f(x) for from the diagrams. Finding the Function Examples Find the function f(x) for from the diagrams. f(x) 1 2 3 1 4 9 4 9 f(x) f(x) f: 0 f: 1 f:2 2 4 5 10 6 11 f(x) = x + 5 f(x) = x2 f(x) = 2x

Illustrating Functions Now try MIA Ex 2.1 Ch10 (page195) 14-Dec-17

Starter Questions Q1. Q2. Find the ratio of cos 60o Q3. 75.9 x 70 Explain why the length a = 36m 30m 24m a 14-Dec-17

Function Notation www.mathsrevision.com Learning Intention S4 Credit Learning Intention Success Criteria To explain the mathematical notation when dealing with functions. Understand function notation. Be able to work with function notation. www.mathsrevision.com 14-Dec-17

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

Function Notation For the function h(x) = 10 – x2. 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

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

Function Notation Now try MIA Ex 3.1 & 3.2 Ch10 (page197) 14-Dec-17

Starter Questions 14-Dec-17

and Quadratic functions Graphs of linear and Quadratic functions S4 Credit Learning Intention Success Criteria To explain the linear and quadratic functions. Understand linear and quadratic functions. Be able to graph linear and quadratic equations. www.mathsrevision.com 14-Dec-17

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

Graph of Quadratic Function 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 It 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 line x = -1 Minimum Turning Point of f(x) is half way between roots  (-1,-4)

Draw the graph of the functions with equations below : 1 2 3 4 5 6 7 8 9 10 -9 -8 -7 -6 -5 -4 -3 -2 -1 -10 x y y = 2x - 5 xy 1 3 y = 2x + 1 xy 1 3 -5 -3 1 x x 1 3 7 x x x x y = x2 xy -2 2 y = 2 - x2 xy -2 2 4 4 -2 2 -2 14-Dec-17

and Quadratic Functions Graphs of Linear and Quadratic Functions Now try MIA Ex 4.1 & 4.2 Ch10 (page 201) 14-Dec-17

Starter Questions Q1. Round to 2 significant figures (a) 52.567 (b) 626 Q2. Why is 2 + 4 x 2 = 10 and not 12 Q3. Solve for x 14-Dec-17

Sketching Quadratic Functions www.mathsrevision.com Learning Intention S4 Credit Learning Intention Success Criteria To show how to sketch quadratic functions. Be able to sketch quadratic functions. www.mathsrevision.com 14-Dec-17

Example 2 : Sketch f(x) = x2 - 7x + 6 Sketching Quadratic Functions We can use a 4 step process to sketch a quadratic function Example 2 : Sketch f(x) = x2 - 7x + 6 Step 1 : Find where the function crosses the x – axis. SAC Method i.e. x2 – 7x + 6 = 0 x - 6 x - 1 (x - 6)(x - 1) = 0 x - 6 = 0 x - 1 = 0 x = 6 (6, 0) x = 1 (1, 0)

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

Now we can sketch the curve y = x2 – 7x + 6 Sketching Quadratic Functions Step 4 : Find where curve cuts y-axis. For x = 0 f(0) = 02 – 7x0 = 6 = 6 (0,6) Now we can sketch the curve y = x2 – 7x + 6 Y Cuts x - axis at 1 and 6 1 6 Cuts y - axis at 6 6 Mini TP (3.5,-2.25) (3.5,-2.25) X

Example 1 : Sketch f(x) = 15 – 2x – x2 Sketching Quadratic Functions We can use a 4 step process to sketch a quadratic function Example 1 : Sketch f(x) = 15 – 2x – x2 Step 1 : Find where the function crosses the x – axis. SAC Method i.e. 15 - 2x - x2 = 0 5 x 3 - x (5 + x)(3 - x) = 0 5 + x = 0 3 - x = 0 x = - 5 (- 5, 0) x = 3 (3, 0)

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

Now we can sketch the curve y = 15 – 2x – x2 Sketching Quadratic Functions Step 4 : Find where curve cuts y-axis. For x = 0 f(0) = 15 – 2x0 – 02 = 15 (0,15) Now we can sketch the curve y = 15 – 2x – x2 Y Cuts x-axis at -5 and 3 -5 3 Cuts y-axis at 15 15 Max TP (-1,16) (-1,16) X

Now try MIA Ex 5.1 Ch10 (page 204) Sketching Quadratic Functions 14-Dec-17

Q2. Multiply out the brackets Starter Questions Q1. Explain why 15% of £80 is £12 Q2. Multiply out the brackets Q3. Q4. 14-Dec-17

The Reciprocal Function S4 Credit Learning Intention Success Criteria To show what the reciprocal function looks like. Know the main points of the reciprocal function. Be able to sketch the reciprocal function. www.mathsrevision.com 14-Dec-17

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

The Reciprocal Function The graph never touches the x or y axis. The axes are said to be asymptotes to the graph The Reciprocal Function The graph has two lines of symmetry at 450 to the axes y x Note that x CANNOT take the value 0.

Draw the graph of the function with equations below : 1 2 3 4 5 6 7 8 9 10 -9 -8 -7 -6 -5 -4 -3 -2 -1 -10 x y y = 1/x xy -10 0.1 10 -1 -0.1 1 -0.1 -1 -10 10 1 0.1 y = 1/(-10) = - 0.1 y = 1/(-1) = - 1 x x y = 1 /(-0.1) = - 10 y = 1 / 0.1 = 10 y = 1 / 1 = 1 x y = 1 / 10 = 0.1 14-Dec-17

Draw the graph of the function with equations below : 1 2 3 4 5 6 7 8 9 10 -9 -8 -7 -6 -5 -4 -3 -2 -1 -10 x y y = 5/x xy -10 1 10 -5 -1 5 -0.5 -1 -5 5 1 0.5 y = 5/(-10) = - 0.5 y = 5/(-5) = - 1 x x y = 5/(-1) = - 5 y = 5 / 1 = 5 x y = 5 / 5 = 1 y = 5 / 10 = 0.5 14-Dec-17

Reciprocal Function Now try MIA Ex 6.1 Ch10 (page 206) 14-Dec-17

Q1. True or false 2a (a – c + 4ab) =2a2 -2ac + 8ab Starter Questions Q1. True or false 2a (a – c + 4ab) =2a2 -2ac + 8ab Q2. Q3. 14-Dec-17

Exponential Function www.mathsrevision.com Learning Intention S4 Credit Learning Intention Success Criteria To show what the exponential function looks like. Know the main points of the exponential function. Be able to sketch the exponential function. www.mathsrevision.com 14-Dec-17

Exponential (to the power of) Graphs Exponential Functions A function in the form f(x) = ax where a > 0, a ≠ 1 is called an exponential function to base a . Consider f(x) = 2x x -3 -2 -1 0 1 2 3 f(x) 1/8 ¼ ½ 1 2 4 8

Draw the graph of the function with equation below : 1 2 3 4 5 6 7 8 9 10 -9 -8 -7 -6 -5 -4 -3 -2 -1 -10 x y y = 2x x = -3 y = 1/8 x = -2 y = 1/4 x = -1 y = 1/2 x x x x x = 0 y = 1 x = 1 y = 2 x = 2 y = 4 x = 3 y = 8 14-Dec-17

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

Exponential Button on the Calculator Remember We can calculate exponential (power) value on the calculator. yx Button looks like 0.111 Examples Calculate the following 2 yx 5 = 32 25 = 3-2 = 3 yx - 8 = 1/9

Exponential Function Now try MIA Ex 7.1 Ch10 (page 208) 14-Dec-17

Starter Questions 39o 14-Dec-17

Summary of Graphs & Functions www.mathsrevision.com Learning Intention S4 Credit Learning Intention Success Criteria To summarise graphs covered in this chapter. Know the main points of the various graphs in this chapter. Be able to identify function and related graphs. www.mathsrevision.com 14-Dec-17

Summary of Graphs & Functions y y x x x x Y Y Reciprocal f(x) = ax + b g(x) = ax2 + bx + c Quadratic Exponential h(x) = a / x Linear k(x) = ax 14-Dec-17

Now try MIA Ex 8.1 Ch10 (page 209) Summary of Graphs & Functions 14-Dec-17

Starter Questions 14-Dec-17

Mathematical Models www.mathsrevision.com Learning Intention S4 Credit Learning Intention Success Criteria To show how we can use functions to model real-life situations. Understand mathematical models using functions Solving problems using mathematical models. www.mathsrevision.com 14-Dec-17

Mathematical Models In real-life scientists look for connections between two or more quantities. 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 other values 14-Dec-17

Mathematical Models Survey Collect data Experiments Tables Organise data Graphs Analyse data Make Predictions 14-Dec-17

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

y = 8x2 y = ax2 y x x x x x x x = 0 y = 0 x = 0.5 y = 2 x = 1.0 y = 8 Does it look like part of a graph we know? x = 0 y = 0 y x = 0.5 y = 2 5 0.5 1.0 1.5 2.0 2.5 10 15 20 25 30 35 40 45 50 x = 1.0 y = 8 x = 1.5 y = 18 x = 2.0 y = 32 y = 8x2 x = 2.5 y = 50 x To find a pick a point of the graph and sub into equation. x x x x (1,8) y = ax2 y = ax2 8 = ax(1)2 x Y y = ax2 a = 8 Pick another point to double check ! (2,32) 32 = 8x(2)2 32 = 8x4 32 = 32

Mathematical Models y = 8x2 Example We can now use the equation to predict other values. y = 8x2 Use the equation to calculate the following : distance when time x = 10 time when distance y = 200 y = 8x2 y = 8x2 y = 8x(10)2 200 = 8x2 y = 8x10 x2 = 25 y = 800 m x = 5 seconds

Mathematical Models Now try MIA Ex 9.1 Ch10 (page 210) 14-Dec-17

Investigation Area & Perimeter How can we check this is correct ? P = x + (8-x) + x + (8-x) P = 16 metres I have 16 metres of fencing in my garage. I want to use it to create a rectangular shaped area in my back garden so that I can grow my own vegetables. What is the maximum area I can enclose with my fence. x (16– 2x) ÷ 2 8-x 8 - x 8 - x ? x 14-Dec-17

Investigation Area & Perimeter x 8 - x 8 - x x Investigate the best way to come up with all the possible rectangular areas that can be made from lengths that are whole numbers with a perimeter of 16 metres 14-Dec-17

Investigation Area & Perimeter x = Area Breadth Length 1 7 7 Now plot Length against Area 2 6 12 3 5 15 4 4 16 Geogebra Link 5 3 15 6 2 12 7 1 7 8 You may need to download Geogebra 14-Dec-17

Exam Type Questions 14-Dec-17

Created by Mr. Lafferty@www.mathsrevision.com Exam Type Questions 14-Dec-17 Created by Mr. Lafferty@www.mathsrevision.com

Exam Type Questions 14-Dec-17

Created by Mr. Lafferty@www.mathsrevision.com Exam Type Questions 14-Dec-17 Created by Mr. Lafferty@www.mathsrevision.com

Exam Type Questions 14-Dec-17

Created by Mr. Lafferty@www.mathsrevision.com Exam Type Questions 14-Dec-17 Created by Mr. Lafferty@www.mathsrevision.com

Created by Mr. Lafferty@www.mathsrevision.com Exam Type Questions 14-Dec-17 Created by Mr. Lafferty@www.mathsrevision.com

Exam Type Questions