Transcript The Laws Of Surds.

Functions
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S4 Credit
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
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Starter Questions
S4 Credit
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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.
17-Jul-15
Created by Mr. [email protected]
Functions
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S4 Credit
Learning Intention
1. To explain what a function
is in terms of a diagram
and formula.
Success Criteria
1. Understand the term
function.
2. Apply knowledge to find
functions given a diagram.
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What are Functions ?
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S4 Credit
Functions describe how one quantity
relates to another
Car
Parts
Assembly
line
Cars
What are Functions ?
S4 Credit
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Functions describe how one quantity
relates to another
Dirty
x
Input
Washing
Machine
Function
f(x)
Clean
y
Output
y = f(x)
Defining a Functions
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S4 Credit
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
S4 Credit
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Functions can be illustrated in a number of ways:
1) by a formula.
2) by arrow diagram.
Example
FORMULA
Suppose that
f(x) = x2 + 3x
Domain
f: A  B
where A = { -3, -2, -1, 0, 1}.
then f(-3) = 0
f(0) = 0
f(-2) = -2
f(1) = 4
f(-1) = -2
is defined by
Range B = {-2, 0, 4}
Illustrating Functions
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S4 Credit
A
ARROW
DIAGRAM
f(x) = x2 + x
B
f(-3)
-3 = 0
f(-2)
-2 = -2
-1 = -2
f(-1)
f(0)
0 =0
1 =4
f(1)
Finding the Function
S4 Credit
Examples
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Find the output or input values for the functions below :
4
12
5
15
6
18
f(x) = 3x
6
36
f: 0
-1
7
49
f: 1
3
8
64
f:2
7
f(x) = 4x - 1
f(x) = x2
Finding the Function
S4 Credit
Examples
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Find the function f(x) for from the diagrams.
f(x)
4
9
5
10
6
11
f(x) = x + 5
1
f(x)
f(x)
1
f: 0
2
4
f: 1
2
3
9
f:2
4
f(x) = x2
f(x) = 2x
0
Illustrating Functions
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S4 Credit
Now try MIA
Ex 2.1
Ch10 (page195)
17-Jul-15
Created by Mr.
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Starter Questions
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S4 Credit
Q1.
1.5% of £500
Q2.
Find the ratio of cos 60o
Q3.
75.9 x 70
Q4.
the length a = 36m
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30m
Explain why
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24m
a
12
Function Notation
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S4 Credit
Learning Intention
1. To explain the mathematical
notation when dealing with
functions.
Success Criteria
1. Understand function notation.
2. Be able to work with function
notation.
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Function Notation
S4 Credit
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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
S4 Credit
Examples
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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
S4 Credit
Examples
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For the function
g(x) = x2 + x
Calculate g(0) , g(3) and h(2a)
g(x) = x2 + x 
g(0) = 02 + 0 = 0
g(3) = 32 + 3 = 12
g(2a) = (2a)2 +2a = 4a2 + 2a
Function Notation
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S4 Credit
Now try MIA
Ex 3.1 & 3.2
Ch10 (page197)
17-Jul-15
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Starter Questions
S4 Credit
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1. Mutliply out 2y( y - 4)
2. Explain why 9x2 - 36 factorises to 9(x - 2)(x + 2)
3. 12% of £22
4. Tidy up the expression
7 - (-10) × 3
17-Jul-15
Created by Mr. Lafferty Maths Dept.
Graphs of linear
and Quadratic functions
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S4 Credit
Learning Intention
1. To explain the linear and
quadratic functions.
Success Criteria
1. Understand linear and
quadratic functions.
2. Be able to graph linear and
quadratic equations.
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Graphs of linear
and Quadratic functions
S4 Credit
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) = ax + b is a linear
function.
Its graph is a straight line
with equation y = ax + b
y
Output (Range)
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A graph gives a picture of a function
b = 0 in this
example !
Input (Domain)
x
A function of the form
f(x) = ax2 + bx +c a ≠ 0
Graph
Quadratic
Function
is calledof
a quadratic
function
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S4 Credit
and its graph is a parabola with
equation y = ax2 + bx + c
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
(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)
Draw the graph of the functions with equations below :
y
10
y = 2x - 5
9
Outcome
2
8
x 0 1 3
y -5 -3 1
y = 2x + 1
7
x 0 1 3
y 1 3 7
6
x
x
5
4
x
x
3
2
1
-10
-9
-8
-7
-6
-5
-4
-3
-2
x
y=2-
x2
x -2 0 2
y -2 2 -2
-1
0
-1
-2
1
2
3
4
5
x
-3
-4
-5
-6
-7
-8
-9
-10
17-Jul-15
Created by Mr. Lafferty Maths Dept
6
7
8
9
10
x
y = x2
x -2 0 2
y 4 0 4
Graphs of Linear
and Quadratic Functions
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S4 Credit
Now try MIA
Ex 4.1 & 4.2
Ch10 (page 201)
17-Jul-15
Created by Mr.
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Starter Questions
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S4 Credit
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
2 x  20  8  x
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24
Sketching
Quadratic Functions
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S4 Credit
Learning Intention
Success Criteria
1. To show how to sketch
quadratic functions.
1. Be able to sketch quadratic
functions.
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Created by Mr. [email protected]
Sketching
S4 Credit
Quadratic Functions
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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
x-6=0
x=6
-1
(x - 6)(x - 1) = 0
(6, 0)
x-1=0
x = 1 (1, 0)
Sketching
Quadratic Functions
S4 Credit
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Step 2 :
(6 + 1) ÷ 2
=3.5
Find equation of axis of symmetry.
It is half way between points in step 1
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 = -6.25
Turning point TP is a Minimum at (3.5, -6.25)
Sketching
Quadratic Functions
S4 Credit
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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
Cuts y - axis at 6
Mini TP (3.5,-6.25)
X
Sketching
S4 Credit
Quadratic Functions
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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
5+x=0
-x
(5 + x)(3 - x) = 0
x = - 5 (- 5, 0)
3-x=0
x = 3 (3, 0)
Sketching
Quadratic Functions
S4 Credit
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Step 2 :
(-5 + 3) ÷ 2
= -1
Step 3 :
Find equation of axis of symmetry.
It is half way between points in step 1
Equation is x = -1
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)
Sketching
Quadratic Functions
S4 Credit
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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
-5 and 33
Cuts y-axis at 15
Max TP (-1,16)
X
Sketching
Quadratic Functions
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S4 Credit
Now try MIA
Ex 5.1
Ch10 (page 204)
17-Jul-15
Created by Mr.
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Starter Questions
S4 Credit
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Q1. Explain why 15% of £80 is £12
Q2. Multiply out the brackets
(a)
2 - 6(x - 3)
Q3. Find g when
(b)
(x - 4)( 3x + 5)
7 - g  32
Q4. If a  2, b  6, c  3
Is the value of ac2  b 2 equal to 0 , -18 or 54
17-Jul-15
Created by Mr. Lafferty Maths Dept.
The Reciprocal Function
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S4 Credit
Learning Intention
1. To show what the
reciprocal function looks
like.
Success Criteria
1. Know the main points of the
reciprocal function.
2. Be able to sketch the
reciprocal function.
17-Jul-15
Created by Mr. [email protected]
The Reciprocal Function
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S4 Credit
y is inversely
proportional
to x
The function of the form
a
f (x ) 
x  0
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
a
y 
x
x  0
The graph never touches
The
Reciprocal
the x or y axis.
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S4 Credit
The axes are said to be
asymptotes to the graph
y
Function
The graph has two
lines of symmetry
at 450 to the axes
x
Note that x CANNOT take the value 0.
Draw the graph of the function with equations below :
y
y = 1/x
10
9
x -10 -1 -0.1 0.1 1 10
y -0.1 -1 -10 10 1 0.1
8
7
6
5
4
y = 1/(-10) = - 0.1
3
2
y = 1/(-1) = - 1
y = 1 /(-0.1) = - 10
x
-10
-9
1
-8
-7
-6
-5
-4
-3
-2
x
-1
-3
-4
-5
-6
y=1/1=1
17-Jul-15
-1
-2
y = 1 / 0.1 = 10
y = 1 / 10 = 0.1
0
-7
-8
x
-9
-10
Created by Mr. Lafferty Maths
Dept
1
2
3
4
5
6
7
8
9
10
x
Draw the graph of the function with equations below :
y
10
y = 5/x
9
8
x -10 -5 -1 1 5 10
y -0.5 -1 -5 5 1 0.5
7
6
5
4
y = 5/(-10) = - 0.5
3
2
y = 5/(-5) = - 1
y = 5/(-1) = - 5
y=5/1=5
x
-10
-9
1
-8
-7
-6
x
-5
-4
-3
-2
-1
-2
-3
x
-4
-5
-6
y=5/5=1
-7
-8
y = 5 / 10 = 0.5
17-Jul-15
0
-1
-9
-10
Created by Mr. Lafferty Maths
Dept
1
2
3
4
5
6
7
8
9
10
x
Reciprocal Function
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S4 Credit
Now try MIA
Ex 6.1
Ch10 (page 206)
17-Jul-15
Created by Mr.
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Starter Questions
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S4 Credit
Q1. True or false 2a (a – c + 4ab) =2a2 -2ac + 8ab
Q2. Find g when
7 - 8g  95
Q3. If a  9, b  7, c  4
Find
ab  c 2
17-Jul-15
Created by Mr. Lafferty Maths Dept.
Exponential Function
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S4 Credit
Learning Intention
1. To show what the exponential
function looks like.
Success Criteria
1. Know the main points of the
exponential function.
2. Be able to sketch the
exponential function.
17-Jul-15
Created by Mr. [email protected]
Exponential
(to the power of) Graphs
S4 Credit
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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
f(x)
-3
-2
-1
0
1
2
3
1/
8
¼
½
1
2
4
8
Draw the graph of the function with equation below :
y
y =
x = -3
2x
10
9
8
y = 1/8
x = -2
y = 1/4
x = -1
y = 1/2
x=0
y=1
x=1
y=2
x=2
y=4
x=3
y=8
7
6
5
4
3
x
x
x
x
2
1
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
-1
-2
-3
-4
-5
-6
-7
-8
-9
-10
17-Jul-15
Created by Mr. Lafferty Maths
Dept
1
2
3
4
5
6
7
8
9
10
x
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S4 Credit
Graph
The graph is like
y = 2x
(0,1)
(1,2)
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
S4 Credit
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Remember
We can calculate exponential (power) value
on the calculator.
Button looks like
Examples
yx
0.111
Calculate the following
25 =
2
yx
5
=
32
3-2 =
3
yx
-
8
=
1/9
Exponential Function
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S4 Credit
Now try MIA
Ex 7.1
Ch10 (page 208)
17-Jul-15
Created by Mr.
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Starter Questions
S4 Credit
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1. If a triangle is right angled and two sides
have length 10 and 9.
What are the possible sizes of the third side.
2. Factorise x2 + 8x + 15
3. The missing angles are 90 and 57. Explain why?
17-Jul-15
Created by Mr. Lafferty Maths Dept.
39o
Summary of
Graphs & Functions
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S4 Credit
Learning Intention
1. To summarise graphs covered
in this chapter.
Success Criteria
1. Know the main points of the
various graphs in this chapter.
2. Be able to identify function
and related graphs.
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Summary of
Graphs & Functions
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S4 Credit
Y
Y
x
x
Exponential h(x) = a / x
17-Jul-15
x
x
Reciprocal
Quadratic
y
y
f(x) = ax + b
g(x) = ax2 + bx + c
Linear
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k(x) = ax
Summary of
Graphs & Functions
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S4 Credit
Now try MIA
Ex 8.1
Ch10 (page 209)
17-Jul-15
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Starter Questions
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S4 Credit
1. Is it true that 16x2 - 36 factorises to
4(2x - 3)(2x + 3)
2. Write down what you understand by the term
(SOH)(C AH)(T OA)
17-Jul-15
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Mathematical Models
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S4 Credit
Learning Intention
Success Criteria
1. To show how we can use
functions to model real-life
situations.
1. Understand mathematical
models using functions
2. Solving problems using
mathematical models.
17-Jul-15
Created by Mr. [email protected]
Mathematical Models
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S4 Credit
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 other values
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Mathematical Models
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S4 Credit
Survey
Collect data
Tables
Organise data
Analyse data
Make Predictions
17-Jul-15
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Experiments
Graphs
Mathematical Models
S4 Credit
Example
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McLaren are testing a new Formula 1 car.
Data was collected and organised into the table below:
Time (x seconds)
0.0 0.5 1.0 1.5 2.0 2.5
Distance ( y metres) 0
2
8 18 32 50
By plotting the data on a graph and analysing the result
is there a connection between the variables
time and distance ?
x=0
y=0
x = 0.5
y=2
50
x = 1.0
y=8
45
x = 1.5
y = 18
x = 2.0
y = 32
35
x = 2.5
y = 50
y
40
To find a pick a point of the
graph and sub into equation.
(1,8)
30
25
20
15
y = 8x2
x
10
5
0x
x
0.5
y=
y =
8 = ax(1)2
a=8
Pick another point to double check !
(2,32)
ax2
Does it look like
part of a graph
we know?
32 = 8x(2)2
32 = 8x4
x
1.0
ax2
32 = 32
1.5
2.0
2.5
x
Y
y = ax2
x
Mathematical Models
S4 Credit
Example
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We can now use the equation to predict other values.
y = 8x2
Use the equation to calculate the following :
distance when time x = 10
y = 8x2
y = 8x(10)2
y = 8x10
y = 800 m
time when distance y = 200
y = 8x2
200 = 8x2
x2 = 25
x = 5 seconds
Mathematical Models
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S4 Credit
Now try MIA
Ex 9.1
Ch10 (page 210)
17-Jul-15
Created by Mr.
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Investigation
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S4 Credit
HowArea
can we &
check
this is correct ?
Perimeter
P metres
= x + (8-x)
+ x + (8-x)
I have 16
of fencing
in my garage.
P = 16 metres
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
8-x
8
?-x
x
17-Jul-15
(16–8-x
2x) ÷ 2
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Investigation
Area & Perimeter
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S4 Credit
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
17-Jul-15
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Investigation
Area & Perimeter
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S4 Credit
x=
17-Jul-15
Length
Breadth
Area
1
7
7
2
6
12
3
5
15
4
4
16
5
3
15
6
2
12
7
1
7
8
0
0
Created by Mr.
[email protected]
Now plot
Length against Area
Geogebra Link
You may need
to download Geogebra
Exam Type Questions
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S4 Credit
17-Jul-15
Created by Mr.
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Exam Type Questions
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S4 Credit
17-Jul-15
Created by Mr.
[email protected]
Exam Type Questions
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S4 Credit
17-Jul-15
Created by Mr.
[email protected]
Exam Type Questions
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S4 Credit
17-Jul-15
Created by Mr.
[email protected]
Exam Type Questions
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S4 Credit
17-Jul-15
Created by Mr.
[email protected]
Exam Type Questions
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S4 Credit
17-Jul-15
Created by Mr.
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Exam Type Questions
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S4 Credit
17-Jul-15
Created by Mr.
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Exam Type Questions
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S4 Credit
Outcome 1