Transcript The Laws Of Surds.

Functions
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Nat 5
Functions & Graphs
Composite Functions
The Quadratic Function
Exam Type Questions
See Quadratic Functions section
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Starter Questions
Nat 5
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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.
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Functions
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Nat 5
Learning Intention
1. We are learning about
functions and their
associated graphs.
22-Jul-15
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.
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What are Functions ?
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Nat 5
Functions describe how one quantity
relates to another
Car
Parts
Assembly
line
Cars
What are Functions ?
Nat 5
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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)
Finding the Function
Nat 5
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
Defining a Functions
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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
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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
Nat 5
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
Nat 5
Examples
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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
Sketching Function
Nat 5
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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.
f(x) =3x + 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
22-Jul-15
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8
9 10
x
3
Sketching Function
Nat 5
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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.
y = x2 - 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
22-Jul-15
Created by Mr. Lafferty Maths Dept
8
9 10
x
-3
-1
6 -2
0
1
3
-3 -2 6
Function & Graphs
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Nat 5
Now try N5 TJ
Ex 12.1 up to Q9
Ch12 (page117)
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Finding the Function
Nat 5
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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
Remember
Finding
the
Function
4 x 4 =16
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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.
k2 + 6 = 22
k2 = 16
k = √16
k = 4 and - 4
k2 + 6
Function & Graphs
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Nat 5
Now try N5 TJ
Ex 12.1 Q10 onwards
Ch12 (page117)
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Starter Questions
Nat 5
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1 . M u t lip ly o u t 2 y ( y - 4 )
2
2.
E x p la in w h y 9 x - 3 6 f a c t o r is e s t o 9 (x - 2 )(x + 2 )
3.
12 % of £ 2 2
4.
T id y u p t h e e x p r e s s io n
7 - (-1 0 ) × 3
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Graphs of linear
and Quadratic functions
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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.
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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)
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A graph gives a picture of a function
c = 0 in this
example !
Input (Domain)
x
Roots
(0, )
x=
a>0
f(x) = x2 + 4x + 3
f(-2) =(-2)2 + 4x(-2) + 3
= -1
Mini. Point
Line of Symmetry
half way
between roots
Evaluating
Graphs
Quadratic Functions
y = ax2 + bx + c
Max. Point
(0, )
x=
a<0
Line of Symmetry
half way
between roots
A function of the form
f(x) = ax2 + bx +c a ≠ 0
Graph
Quadratic
Function
is calledof
a quadratic
function
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Nat 5
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
Its 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)
Sketching
Quadratic Functions
Nat 5
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Example : Sketch f(x) = x2
{ -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
-4
-5
-3
-2
-1
0
1
2
3
4
5
-1
This function has
one root.
What is it ?
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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 ?
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Sketching
Nat 5
Quadratic Functions
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Example : Sketch f(x) = 4x – x2
{ -1 ≤ x ≤ 5 }
Make a table
x -1
y -5
0
1
2
3
0
3
4
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
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(2,4)
-6
-7
x = 0 and 4
x
-9
-10
What is the
maximum turning
point ?
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Function & Graphs
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Nat 5
Now try N5 TJ
Ex 12.2
Ch12 (page120)
22-Jul-15
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[email protected]