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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 𝒙 and π’š – intercepts for a range of graphs of functions.

    Find out where the following graphs cross the π‘₯-axis and the 𝑦-axis:

    (a) 𝑦 = 4π‘₯ + 8 (b) 𝑦 = 1

    4 π‘₯ βˆ’ 3 (c) 3π‘₯ + 5𝑦– 15 = 0

    (d) 𝑦 = π‘₯Β² βˆ’ 3π‘₯ (e) 𝑦 = π‘₯Β² βˆ’ 16 (f) 𝑦 = π‘₯Β² + 6π‘₯ βˆ’ 27

    (g) 𝑦 = 2π‘₯Β² βˆ’ 18 (h) 𝑦 = 2π‘₯Β² + 5π‘₯ βˆ’ 3

    R2 I can complete the square for a quadratic with coefficient of π’™πŸ = ±𝟏.

    (a) π‘₯2 + 2π‘₯ + 5 (b) 𝑑2 βˆ’ 10𝑑 + 2 (c) 𝑣2 βˆ’ 2𝑣 + 7

    (d) 7 βˆ’ 2π‘₯ βˆ’ π‘₯2 (e) 1 βˆ’ 4t βˆ’ t2 (f) 1 + 2π‘₯ βˆ’ π‘₯2

    R3 I have had experience of graphing linear and quadratic functions.

    1. Sketch the graphs of the following straight lines:

    (a) 𝑦 = 2π‘₯ + 3 (b) 𝑦 = βˆ’3π‘₯ – 2

    (c) 𝑦 = 1

    2 π‘₯ + 1 (d) 2π‘₯ + 𝑦 – 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 it’s nature

    ο‚· Sketch the graph

    (a) 𝑦 = π‘₯Β² βˆ’ 4π‘₯ + 3 (b) 𝑦 = π‘₯Β² βˆ’ 4π‘₯ βˆ’ 12

  • Functions and graphs

    Calderglen High School - Mathematics Department

    2

    3. For the following Quadratic Functions:

    ο‚· Express in the form 𝑦 = π‘Ž(π‘₯ + 𝑏)Β² + 𝑐 ο‚· Find the Turning Point, and state it’s nature, and find where the

    graph cuts the 𝑦-axis.

    (a) 𝑦 = π‘₯Β² + 6π‘₯ βˆ’ 1 (b) 𝑦 = π‘₯Β² βˆ’ 4π‘₯ + 5

    (c) 𝑦 = π‘₯Β² + 3π‘₯ + 4 (d) 𝑦 = π‘₯Β² βˆ’ 5π‘₯ βˆ’ 5

    Section B - Assessment Standard Section

    This section will help you practise for your Assessment Standard Test (Expressions

    and Functions 1.3)

    1. The diagram shows the graph of 𝑦 = 𝑓(π‘₯) with a minimum turning point at

    (βˆ’2, βˆ’2) and a maximum turning point at (2, 3).

    Sketch the graph of 𝑦 = 𝑓(π‘₯ βˆ’ 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 𝑦 = 𝑓(π‘₯) with a maximum turning point at

    (βˆ’4, 8) and a minimum turning point at (0, 0).

    Sketch the graph of 𝑦 = 𝑓(π‘₯ + 2) βˆ’ 3.

    3. The diagram shows the graph of 𝑦 = π‘™π‘œπ‘”π‘(π‘₯ βˆ’ π‘Ž)

    Determine the values of π‘Ž and 𝑏.

    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 𝑦 = π‘™π‘œπ‘”π‘(π‘₯ + π‘Ž)

    Determine the values of π‘Ž and 𝑏.

    5. Sketch the graph of 𝑦 = π‘Žπ‘π‘œπ‘ (π‘₯ βˆ’ πœ‹

    3 ) for 0 ≀ π‘₯ ≀ 2πœ‹ and π‘Ž > 0, clearly

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

    6. Sketch the graph of 𝑦 = π‘Žπ‘ π‘–π‘›(π‘₯ βˆ’ πœ‹

    6 ) for 0 ≀ π‘₯ ≀ 2πœ‹ and π‘Ž > 0, clearly

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

    7. The diagram below shows the graph of 𝑦 = acos(𝑏π‘₯) + 𝑐.

    Write down the values of π‘Ž, 𝑏 and 𝑐.

    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 𝑦 = asin(𝑏π‘₯) + 𝑐.

    Write down the values of π‘Ž, 𝑏 and 𝑐.

    9. The functions 𝑓 and 𝑔, defined on suitable domains, are given

    by 𝑓(π‘₯) = 2π‘₯ + 3 and 𝑔(π‘₯) = π‘₯2+25

    π‘₯2βˆ’25 where π‘₯ β‰  Β±5.

    A third function β„Ž(π‘₯) is defined as β„Ž(π‘₯) = 𝑔(𝑓(π‘₯)).

    (a) Find an expression for β„Ž(π‘₯).

    (b) For which real values of π‘₯ is the function β„Ž(π‘₯) undefined?

    10. A function is given by 𝑓(π‘₯) = 3π‘₯Β² + 1. Find the inverse function π‘“βˆ’1(π‘₯).

    -Ο€/4 Ο€/4 Ο€/2 3Ο€/4 Ο€

    -1

    1

    2

    3

    4

    5

    x

    y

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    Calderglen High School - Mathematics Department

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    Section C – 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 ∈:

    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 ∈ { 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 ∈ 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 } ∩ { 2,3,4,5 }

    (b) { 1,2,3 } ∩ { 3,1,2 }

    (c) βˆ… ∩ { 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 }

    (a) Find A ∩ B, B ∩ C and A ∩ C.

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

    Find A ∩ B ∩ C.

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

    (a) 2 ∈ A (b) 1 βŠ‚ A (c) {1} βŠ‚ A

    (d) 0 ∈ βˆ… (e) A βŠ‚ A (f) 1 ∈ 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,…,12 }.

    List the members of the following sets:

    (a) P ∩ Q (b) P βˆͺ Q (c) P’

    (d) Q’ (e) (P ∩ Q)’ (f) (P βˆͺ Q)’

    (g) P ∩ Q’ (h) P’ ∩ Q (i) P ∩ βˆ…

    O2 I have investigated domains and ranges.

    1. State a suitable domain for the following functions:

    (a) 𝑓(π‘₯) = π‘₯2

    π‘₯βˆ’1 (b) 𝑓(π‘₯) =

    4π‘₯ βˆ’ 2

    2π‘₯ βˆ’ 3 (c) 𝑓(π‘₯) =

    2π‘₯ + 7

    π‘₯2βˆ’ 16

    (d) 𝑓(π‘₯) = π‘₯2βˆ’ 5π‘₯ + 4

    π‘₯2+ 8π‘₯+12 (e) 𝑓(π‘₯) = √10 βˆ’ π‘₯ (f) 𝑓(π‘₯) = √π‘₯2 + 3π‘₯

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

    (a) 𝑓(π‘₯) = 3π‘₯ βˆ’ 4 ; π‘₯ ∈ { 2, 3, 4, 5 }

    (b) 𝑓(π‘₯) = π‘₯2 βˆ’ 3π‘₯ + 4 ; π‘₯ ∈ { βˆ’2, βˆ’1, 0, 1, 2 }

    O3 I can determine a composite function.

    1. Given 𝑓(π‘₯) = 2π‘₯ βˆ’ 3, 𝑔(π‘₯) = π‘₯Β² and β„Ž(π‘₯) = π‘₯Β² + 4, find the following

    functions:

    (a) 𝑓(𝑔(π‘₯)) (b) 𝑔(𝑓(π‘₯)) (c) β„Ž(𝑓(π‘₯))

    (d) 𝑓(𝑓(π‘₯)) (e) 𝑔(β„Ž(π‘₯)) (f) β„Ž(β„Ž(π‘₯))

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    2. Given 𝑓(π‘₯) = π‘₯ βˆ’ 2, 𝑔(π‘₯) = 2

    π‘₯2 and β„Ž(π‘₯) =

    4

    π‘₯+1 , find the following

    functions:

    (a) β„Ž(𝑓(π‘₯)) (b) 𝑔(𝑓(π‘₯)) (c) 𝑓(β„Ž(π‘₯))

    (d) 𝑓(𝑔(π‘₯)) (e) 𝑔(β„Ž(π‘₯)) (f) β„Ž(β„Ž(π‘₯))

    3. Given 𝑓(π‘₯) = π‘₯ + 2, 𝑔(π‘₯) = 𝑒π‘₯ and β„Ž(π‘₯) = tan π‘₯, find the following

    functions:

    (a) 𝑔(𝑓(π‘₯)) (b) 𝑔(𝑔(π‘₯)) (c) β„Ž(𝑓(π‘₯))

    4. Given 𝑓(π‘₯) = 3π‘₯2 + 2π‘₯ βˆ’ 1, 𝑔(π‘₯) = sin π‘₯ and β„Ž(π‘₯) = log4 π‘₯, find the

    following functions:

    (a) 𝑓(𝑔(π‘₯)) (b) β„Ž(𝑓(π‘₯)) (c) 𝑔(𝑔(π‘₯))

    5. Two functions f and g, are defined by 𝑓(π‘₯) = 2π‘₯ + 3 and 𝑔(π‘₯) = 2π‘₯ – 3,

    where x is a real number.

    (a) Find expressions for 𝑓(𝑔(π‘₯)) and 𝑔(𝑓(π‘₯)).

    (b) Determine the least possible value of the product 𝑓(𝑔(π‘₯)) Γ— 𝑔(𝑓(π‘₯)).

    6. Functions 𝑓(π‘₯) = 3π‘₯ βˆ’ 1 and 𝑔(π‘₯) = π‘₯Β² + 7, are defined on a set of real

    numbers.

    (a) Find h(x) where β„Ž(π‘₯)