Functions

1
Functions
Functions Graphs
Composite Functions
The Quadratic Function
Exam Type Questions See Quadratic Functions
section
2
Starter Questions
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.
3
Functions
Nat 5
Learning Intention
Success Criteria

1. Understand the term function.

1. We are learning about functions and their
   associated graphs.

1. Know that the input is the x-coordinate and the
   output is the y-coordinate.

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3. Recognise the graph of a linear and quadratic
function.
4
What are Functions ?
Functions describe how one quantity relates to
another
Car Parts
Cars
Assembly line
5
What are Functions ?
Functions describe how one quantity relates to
another
Dirty
Clean
Washing Machine
y f(x)
y
x

Function
Output
Input
f(x)
6
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 f2
-1 3 7

5
15


6
18

f(x) x2
f(x) 4x - 1
f(x) 3x
7
Defining a Functions
A function can be thought of as the relationship
between Set A (INPUT - the x-coordinate) and
SET B the y-coordinate (Output) .
8
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)
5 is the value of f at 1
1 4
5
f(a)
a 4 is the value of f at a
a 4
9
Function Notation
Examples
For the function h(x) 10 x2. Calculate
h(1) , h(-3) and h(5)
h(1)
h(x) 10 x2 ?
10 12
9
h(-3)
10 (-3)2
10 9 1
h(5)
10 52
10 25 -15
10
Function Notation
Examples
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
11
Sketching Function
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.
12
f(x) 3x 1
Straight Line Functions
x y
0
1
2
3
-1
1
4
7
10
-2
13
Sketching Function
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.
14
y x2 - 3
Quadratic Functions
x y
-1
0
1
3
-3
-2
-3
-2
6
6
Demo
15
Function Graphs
Now try N5 TJ Ex 12.1 up to Q9 Ch12 (page117)
16
Finding the Function
Example
Consider the function f(x) x - 4
(a) Find an expression for f(3a)
( ) - 4
3a
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
17
Finding the Function
Remember 4 x 4 16 Also (-4)x(-4) 16
Example
Consider the function f(x) x2 6
(a) Write down the value of f(k)
k2 6
(b) If f(k) 22 , set up an equation and solve
for k.
k2 6 22
k2 16
k v16
k 4 and - 4
18
Function Graphs
Now try N5 TJ Ex 12.1 Q10 onwards Ch12 (page117)
19
Starter Questions
20
Graphs of linear and Quadratic functions
Nat 5
Learning Intention
Success Criteria

1. Understand linear and quadratic functions.

1. We are learning about linear and quadratic
   functions.

1. Be able to graph linear and quadratic equations.

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21
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 )
A function of the form f(x) mx c is a linear
function.
c 0 in this example !
Output (Range)
Its graph is a straight line with equation y mx
c
Input (Domain)
22
Roots
f(x) x2 4x 3 f(-2) (-2)2 4x(-2) 3
-1
(0, )
a gt 0
Mini. Point
x
Line of Symmetry half way between roots
Evaluating
Graphs
Quadratic Functions y ax2 bx c
c
c
Max. Point
(0, )
a lt 0
x
Line of Symmetry half way between roots
23
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
Its 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)

24
Sketching Quadratic Functions
Example Sketch f(x) x2 -3 x 3
Make a table
x -3 -2 -1 0 1 2 3
y
9
4
1
0
1
4
9
25
What is the equation of symmetry ?
x
x
x 0
x
x
x
This function has one root. What is it ?
(0,0)
What is the minimum turning point ?
x 0
26
Sketching Quadratic Functions
Example Sketch f(x) 4x x2 -1 x
5
Make a table
x -1 0 1 2 3 4 5
y
-5
-5
0
3
4
3
0
27
What is the equation of symmetry ?
x 2
x
x
x
x
x
What are the roots of the function ?
(2,4)
x
x
What is the maximum turning point ?
x 0 and 4
28
Function Graphs
Now try N5 TJ Ex 12.2 Ch12 (page120)