www.mathsrevision.com nat 5 functions functions & graphs composite functions exam type...
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Nat 5
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Functions
Functions & Graphs
Composite Functions
Exam Type QuestionsSee Quadratic Functions section
The Quadratic Function
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Nat 5
Apr 19, 2023Apr 19, 2023 Created by Mr. Created by Mr. [email protected]@mathsrevision.com
Starter QuestionsStarter 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.
19 Apr 202319 Apr 2023 Created by Mr. [email protected] by Mr. [email protected]
Learning IntentionLearning Intention Success CriteriaSuccess Criteria
1.1. Understand the term Understand the term function.function.
1. We are learning about functions and their associated graphs.
2.2. Know that the input is the Know that the input is the x-coordinate and the output x-coordinate and the output is the y-coordinate.is the y-coordinate.
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FunctionsNat 5
3. 3. Recognise the graph of a Recognise the graph of a linear and quadratic linear and quadratic function.function.
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Nat 5
What are Functions ?
Functions describe how one quantity
relates to another
Car Part
s
Assembly line
Cars
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Nat 5
What are Functions ?
Functions describe how one quantity
relates to another
Dirty
Washing Machine
Clean
OutputInputyx
Functionf(x)
y = f(x)
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Nat 5
Finding the Function
Find the output or input values for the functions below :
6
7
8
36
49
64
f(x) = x2
f: 0
f: 1
f:2
-1
3
7f(x) = 4x - 1
4 12
f(x) = 3x
5 15
6 18
Examples
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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) .
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Nat 5
The standard way to represent a function
is by a formula.
Function Notation
Examplef(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
f(a) = a + 4 is the value of f at a
1 + 4 = 5
a + 4
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Nat 5
For the function h(x) = 10 – x2.
Calculate h(1) , h(-3) and h(5)
h(1) =
Examples
h(-3) = h(5) =
h(x) = 10 – x2
Function Notation
10 – 12 = 9
10 – (-3)2 =
10 – 9 = 1
10 – 52 =
10 – 25 = -15
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Nat 5
For the function g(x) = x2 + x
Calculate g(0) , g(3) and g(2a)
g(0) =
Examples
g(3) = g(2a) =
g(x) = x2 + x
Function Notation
02 + 0 =
0
32 + 3 =
12
(2a)2 +2a =
4a2 + 2a
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Nat 5
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.
Sketching Function
19 Apr 2023 Created by Mr. Lafferty Maths Dept
0 1 2 3 4 5 6 7 8 9 10-9 -8 -7 -6 -5 -4 -3 -2 -1-10 x
y
1
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10
-1
-2
-3
-4
-5
-6
-7
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-9
-10
f(x) =3x + 1
x
y
0
1
1
4
2
7
Straight Line Functions3
10
-1
-2
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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.
Sketching Function
19 Apr 2023 Created by Mr. Lafferty Maths Dept
0 1 2 3 4 5 6 7 8 9 10-9 -8 -7 -6 -5 -4 -3 -2 -1-10 x
y
1
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-1
-2
-3
-4
-5
-6
-7
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-9
-10
y = x2 - 3
x
y
-1
-2
0
-3
1
-2
Quadratic Functions3
6
-3
6
Demo
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Nat 5
19 Apr 202319 Apr 2023 Created by Mr. Created by Mr. [email protected]@www.mathsrevision.com
Now try N5 TJ
Ex 12.1 up to Q9
Ch12 (page117)
Function & Graphs
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Nat 5
Finding the Function
Example :Consider the function f(x) = x - 4
(a) Find an expression for f(3a) ( ) - 43a 3a - 4
(b) Find an expression for f(2p)
Example :Consider the function f(x) = 3x2 + 2
3( )2 + 22p
3(4p2) + 2
12p2 + 2
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Nat 5
Finding the Function
Example :Consider the function f(x) = x2 + 6
(b) If f(k) = 22 ,
set up an equation and solve for k.
(a) Write down the value of f(k)
k2 + 6 = 22k2 = 16k = √16k = 4 and - 4
k2 + 6
Remember
4 x 4 =16
Also
(-4)x(-4) = 16
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19 Apr 202319 Apr 2023 Created by Mr. Created by Mr. [email protected]@www.mathsrevision.com
Now try N5 TJ
Ex 12.1 Q10 onwards
Ch12 (page117)
Function & Graphs
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Nat 5
19 Apr 202319 Apr 2023 Created by Mr. Lafferty Maths Dept.Created by Mr. Lafferty Maths Dept.
Starter Questions
2
1. Mutliply out 2y( y - 4)
2. Explain why 9x - 36 f actorises to 9(x - 2)(x +2)
3. 12% of £ 22
4. Tidy up the expression
7 - (-10) × 3
19 Apr 202319 Apr 2023 Created by Mr. [email protected] by Mr. [email protected]
Learning IntentionLearning Intention Success CriteriaSuccess Criteria
1.1. Understand linear and Understand linear and quadratic functions.quadratic functions.
1. We are learning about linear and quadratic functions.
2.2. Be able to graph linear and Be able to graph linear and quadratic equations.quadratic equations.
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Nat 5
Graphs of linear
and Quadratic functions
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Nat 5
A graph gives a picture of a functionIt 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) = mx + c is a linear function.
Its graph is a straight line with equation y = mx + c
y
xInput (Domain)
Outp
ut
(Rang
e)
c = 0 in this example !
Quadratic Functions
y = ax2 + bx + c
Graphs
Evaluating
Roots
Mini. Point
(0, )
(0, )
Max. Point
Line of Symmetryhalf way
between roots
Line of Symmetryhalf way
between roots
a > 0
a < 0
f(x) = x2 + 4x + 3f(-2) =(-2)2 + 4x(-2) + 3 = -1
x=
x=
cc
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Its equation is y = x2 + 2x - 3
The parabola shown here is the graph of the function f defined by f(x) = x2 + 2x - 3
Graph of Quadratic Function
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)
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
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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
Outcome 2
0 1 2 3 4 5 6 7 8 9 10-9 -8 -7 -6 -5 -4 -3 -2 -1-10 x
y
1
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-1
-2
-3
-4
-5
-6
-7
-8
-9
-10
x
19 Apr 2023 Created by Mr. Lafferty Maths Dept
x x
x x
What is the equation of symmetry ?
x = 0
What is the minimum turning
point ?
(0,0)This function has
one root.
What is it ?
x = 0
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Nat 5
Sketching
Quadratic Functions
Example : Sketch f(x) = 4x – x2
{ -1 ≤ x ≤ 5 }
Make a table
x -1 0 1 2 3 4 5
y 0 3 4 3 0-5 -5
Outcome 2
0 1 2 3 4 5 6 7 8 9 10-9 -8 -7 -6 -5 -4 -3 -2 -1-10 x
y
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-1
-2
-3
-4
-5
-6
-7
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-9
-10
x
19 Apr 2023 Created by Mr. Lafferty Maths Dept
x
x
x
x
What is the equation of symmetry ?
x = 2
What is the maximum turning
point ?
(2,4)
x
x
What are the roots of the function ?
x = 0 and 4
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19 Apr 202319 Apr 2023 Created by Mr. Created by Mr. [email protected]@www.mathsrevision.com
Now try N5 TJ
Ex 12.2
Ch12 (page120)
Function & Graphs