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• Higher Higher Portfolio Functions and Graphs

EF3. Functions and Graphs

Section A - Revision Section

This section will help you revise previous learning which is required in this topic.

R1 I have investigated π and π β intercepts for a range of graphs of functions.

Find out where the following graphs cross the π₯-axis and the π¦-axis:

(a) π¦ = 4π₯ + 8 (b) π¦ = 1

4 π₯ β 3 (c) 3π₯ + 5π¦β 15 = 0

(d) π¦ = π₯Β² β 3π₯ (e) π¦ = π₯Β² β 16 (f) π¦ = π₯Β² + 6π₯ β 27

(g) π¦ = 2π₯Β² β 18 (h) π¦ = 2π₯Β² + 5π₯ β 3

R2 I can complete the square for a quadratic with coefficient of ππ = Β±π.

(a) π₯2 + 2π₯ + 5 (b) π‘2 β 10π‘ + 2 (c) π£2 β 2π£ + 7

(d) 7 β 2π₯ β π₯2 (e) 1 β 4t β t2 (f) 1 + 2π₯ β π₯2

1. Sketch the graphs of the following straight lines:

(a) π¦ = 2π₯ + 3 (b) π¦ = β3π₯ β 2

(c) π¦ = 1

2 π₯ + 1 (d) 2π₯ + π¦ β 4 = 0

2. For the following Quadratic Functions:

ο· Calculate where the graph crosses the x-axis and the y-axis

ο· Find the Turning Point and state itβs nature

ο· Sketch the graph

(a) π¦ = π₯Β² β 4π₯ + 3 (b) π¦ = π₯Β² β 4π₯ β 12

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Calderglen High School - Mathematics Department

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3. For the following Quadratic Functions:

ο· Express in the form π¦ = π(π₯ + π)Β² + π ο· Find the Turning Point, and state itβs nature, and find where the

graph cuts the π¦-axis.

(a) π¦ = π₯Β² + 6π₯ β 1 (b) π¦ = π₯Β² β 4π₯ + 5

(c) π¦ = π₯Β² + 3π₯ + 4 (d) π¦ = π₯Β² β 5π₯ β 5

Section B - Assessment Standard Section

and Functions 1.3)

1. The diagram shows the graph of π¦ = π(π₯) with a minimum turning point at

(β2, β2) and a maximum turning point at (2, 3).

Sketch the graph of π¦ = π(π₯ β 3) + 2.

x

y

(0, 1)

(2, 3)

y = f(x)

(-2, -2)

0

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Calderglen High School - Mathematics Department

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2. The diagram shows the graph of π¦ = π(π₯) with a maximum turning point at

(β4, 8) and a minimum turning point at (0, 0).

Sketch the graph of π¦ = π(π₯ + 2) β 3.

3. The diagram shows the graph of π¦ = ππππ(π₯ β π)

Determine the values of π and π.

x

y

(2, 8)

y = f(x)

(-4, 8)

0

x

y

(4, 0)

(8, 1)

0

1

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Calderglen High School - Mathematics Department

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4. The diagram shows the graph of π¦ = ππππ(π₯ + π)

Determine the values of π and π.

5. Sketch the graph of π¦ = ππππ (π₯ β π

3 ) for 0 β€ π₯ β€ 2π and π > 0, clearly

showing the maximum and minimum values and where it cuts the π₯-axis.

6. Sketch the graph of π¦ = ππ ππ(π₯ β π

6 ) for 0 β€ π₯ β€ 2π and π > 0, clearly

showing the maximum and minimum values and where it cuts the π₯-axis.

7. The diagram below shows the graph of π¦ = acos(ππ₯) + π.

Write down the values of π, π and π.

1

x

y

(5, 0)

(7, 1)

0

Ο/3 2Ο/3 Ο

-5

-4

-3

-2

-1

1

2

3

x

y

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Calderglen High School - Mathematics Department

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8. The diagram below shows the graph of π¦ = asin(ππ₯) + π.

Write down the values of π, π and π.

9. The functions π and π, defined on suitable domains, are given

by π(π₯) = 2π₯ + 3 and π(π₯) = π₯2+25

π₯2β25 where π₯ β  Β±5.

A third function β(π₯) is defined as β(π₯) = π(π(π₯)).

(a) Find an expression for β(π₯).

(b) For which real values of π₯ is the function β(π₯) undefined?

10. A function is given by π(π₯) = 3π₯Β² + 1. Find the inverse function πβ1(π₯).

-Ο/4 Ο/4 Ο/2 3Ο/4 Ο

-1

1

2

3

4

5

x

y

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Section C β Operational Skills Section

This section provides problems with the operational skills associated with

Functions and Graphs.

O1 I can understand and use basic set notation.

1. Using the { } brackets notation, list the following sets:

(a) The set of the first ten prime numbers.

(b) The set of odd numbers greater than 20 but less than 30.

2. Describe the following sets in words:

(a) { Cone, Pyramid }

(b) { 1, 4, 9, 16, 25 }

3. Connect these numbers with the appropriate set, using β:

Numbers: -3 , 0 , 5

2 ο­ , 7

Sets: N, W, Z , Q

4. State which of the following are true and which are false:

(a) 2 β { prime numbers }

(b) { 0 } is the empty set

(c) { k,l,m,n } = { m,l,k,n }

(d) If A = { whole numbers greater than 50 }, then 46 β A

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5. Using set notation, rewrite the following:

(a) 3 is a member of the set W.

(b) The empty set.

(c) x does not belong to the set A.

(d) S is a subset of the set T.

(e) The set P is equal to the set Q.

6. S = { 1,2,3,4,5,6,7,8,9,10 }. List the following subsets of S:

(a) The set of prime numbers in S.

(b) The set of elements in S which are factors of 70.

7. Find a set equal to each of the following:

(a) { 1,2,3 } β© { 2,3,4,5 }

(b) { 1,2,3 } β© { 3,1,2 }

(c) β β© { 2,3,4,5 }

8. E = { 1,2,3,4,5,6,8,10 } A = { 1,2,3,4 } B = { 3,4,5 } and C = { 2,4,6,8,10 }

(b) The set of elements common to A,B and C is denoted by A β© B β© C.

9. Given that A = { 0,1,2 }, which of the following are true?

(a) 2 β A (b) 1 β A (c) {1} β A

(d) 0 β β (e) A β A (f) 1 β A

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Calderglen High School - Mathematics Department

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10. P = { 1,2,3,4,5,6,7 } Q = { 5,6,7,8,9,10 } are subsets of E = { 1,2,3,β¦,12 }.

List the members of the following sets:

(a) P β© Q (b) P βͺ Q (c) Pβ

(d) Qβ (e) (P β© Q)β (f) (P βͺ Q)β

O2 I have investigated domains and ranges.

1. State a suitable domain for the following functions:

(a) π(π₯) = π₯2

π₯β1 (b) π(π₯) =

4π₯ β 2

2π₯ β 3 (c) π(π₯) =

2π₯ + 7

π₯2β 16

(d) π(π₯) = π₯2β 5π₯ + 4

π₯2+ 8π₯+12 (e) π(π₯) = β10 β π₯ (f) π(π₯) = βπ₯2 + 3π₯

2. State the range of each function given its domain:

(a) π(π₯) = 3π₯ β 4 ; π₯ β { 2, 3, 4, 5 }

(b) π(π₯) = π₯2 β 3π₯ + 4 ; π₯ β { β2, β1, 0, 1, 2 }

O3 I can determine a composite function.

1. Given π(π₯) = 2π₯ β 3, π(π₯) = π₯Β² and β(π₯) = π₯Β² + 4, find the following

functions:

(a) π(π(π₯)) (b) π(π(π₯)) (c) β(π(π₯))

(d) π(π(π₯)) (e) π(β(π₯)) (f) β(β(π₯))

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Calderglen High School - Mathematics Department

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2. Given π(π₯) = π₯ β 2, π(π₯) = 2

π₯2 and β(π₯) =

4

π₯+1 , find the following

functions:

(a) β(π(π₯)) (b) π(π(π₯)) (c) π(β(π₯))

(d) π(π(π₯)) (e) π(β(π₯)) (f) β(β(π₯))

3. Given π(π₯) = π₯ + 2, π(π₯) = ππ₯ and β(π₯) = tan π₯, find the following

functions:

(a) π(π(π₯)) (b) π(π(π₯)) (c) β(π(π₯))

4. Given π(π₯) = 3π₯2 + 2π₯ β 1, π(π₯) = sin π₯ and β(π₯) = log4 π₯, find the

following functions:

(a) π(π(π₯)) (b) β(π(π₯)) (c) π(π(π₯))

5. Two functions f and g, are defined by π(π₯) = 2π₯ + 3 and π(π₯) = 2π₯ β 3,

where x is a real number.

(a) Find expressions for π(π(π₯)) and π(π(π₯)).

(b) Determine the least possible value of the product π(π(π₯)) Γ π(π(π₯)).

6. Functions π(π₯) = 3π₯ β 1 and π(π₯) = π₯Β² + 7, are defined on a set of real

numbers.

(a) Find h(x) where β(π₯)

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