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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 𝒚 – intercepts for a range of graphs of functions.

Find out where the following graphs cross the 𝑥-axis and the 𝑦-axis:

(a) 𝑦 = 4𝑥 + 8 (b) 𝑦 = 1

4 𝑥 − 3 (c) 3𝑥 + 5𝑦– 15 = 0

(d) 𝑦 = 𝑥² − 3𝑥 (e) 𝑦 = 𝑥² − 16 (f) 𝑦 = 𝑥² + 6𝑥 − 27

(g) 𝑦 = 2𝑥² − 18 (h) 𝑦 = 2𝑥² + 5𝑥 − 3

R2 I can complete the square for a quadratic with coefficient of 𝒙𝟐 = ±𝟏.

(a) 𝑥2 + 2𝑥 + 5 (b) 𝑡2 − 10𝑡 + 2 (c) 𝑣2 − 2𝑣 + 7

(d) 7 − 2𝑥 − 𝑥2 (e) 1 − 4t − t2 (f) 1 + 2𝑥 − 𝑥2

1. Sketch the graphs of the following straight lines:

(a) 𝑦 = 2𝑥 + 3 (b) 𝑦 = −3𝑥 – 2

(c) 𝑦 = 1

2 𝑥 + 1 (d) 2𝑥 + 𝑦 – 4 = 0

2. For the following Quadratic Functions:

 Calculate where the graph crosses the x-axis and the y-axis

 Find the Turning Point and state it’s nature

 Sketch the graph

(a) 𝑦 = 𝑥² − 4𝑥 + 3 (b) 𝑦 = 𝑥² − 4𝑥 − 12

• Functions and graphs

Calderglen High School - Mathematics Department

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3. For the following Quadratic Functions:

 Express in the form 𝑦 = 𝑎(𝑥 + 𝑏)² + 𝑐  Find the Turning Point, and state it’s nature, and find where the

graph cuts the 𝑦-axis.

(a) 𝑦 = 𝑥² + 6𝑥 − 1 (b) 𝑦 = 𝑥² − 4𝑥 + 5

(c) 𝑦 = 𝑥² + 3𝑥 + 4 (d) 𝑦 = 𝑥² − 5𝑥 − 5

Section B - Assessment Standard Section

and Functions 1.3)

1. The diagram shows the graph of 𝑦 = 𝑓(𝑥) with a minimum turning point at

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

Sketch the graph of 𝑦 = 𝑓(𝑥 − 3) + 2.

x

y

(0, 1)

(2, 3)

y = f(x)

(-2, -2)

0

• Functions and graphs

Calderglen High School - Mathematics Department

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2. The diagram shows the graph of 𝑦 = 𝑓(𝑥) with a maximum turning point at

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

Sketch the graph of 𝑦 = 𝑓(𝑥 + 2) − 3.

3. The diagram shows the graph of 𝑦 = 𝑙𝑜𝑔𝑏(𝑥 − 𝑎)

Determine the values of 𝑎 and 𝑏.

x

y

(2, 8)

y = f(x)

(-4, 8)

0

x

y

(4, 0)

(8, 1)

0

1

• Functions and graphs

Calderglen High School - Mathematics Department

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4. The diagram shows the graph of 𝑦 = 𝑙𝑜𝑔𝑏(𝑥 + 𝑎)

Determine the values of 𝑎 and 𝑏.

5. Sketch the graph of 𝑦 = 𝑎𝑐𝑜𝑠(𝑥 − 𝜋

3 ) for 0 ≤ 𝑥 ≤ 2𝜋 and 𝑎 > 0, clearly

showing the maximum and minimum values and where it cuts the 𝑥-axis.

6. Sketch the graph of 𝑦 = 𝑎𝑠𝑖𝑛(𝑥 − 𝜋

6 ) for 0 ≤ 𝑥 ≤ 2𝜋 and 𝑎 > 0, clearly

showing the maximum and minimum values and where it cuts the 𝑥-axis.

7. The diagram below shows the graph of 𝑦 = acos(𝑏𝑥) + 𝑐.

Write down the values of 𝑎, 𝑏 and 𝑐.

1

x

y

(5, 0)

(7, 1)

0

π/3 2π/3 π

-5

-4

-3

-2

-1

1

2

3

x

y

• Functions and graphs

Calderglen High School - Mathematics Department

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8. The diagram below shows the graph of 𝑦 = asin(𝑏𝑥) + 𝑐.

Write down the values of 𝑎, 𝑏 and 𝑐.

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

by 𝑓(𝑥) = 2𝑥 + 3 and 𝑔(𝑥) = 𝑥2+25

𝑥2−25 where 𝑥 ≠ ±5.

A third function ℎ(𝑥) is defined as ℎ(𝑥) = 𝑔(𝑓(𝑥)).

(a) Find an expression for ℎ(𝑥).

(b) For which real values of 𝑥 is the function ℎ(𝑥) undefined?

10. A function is given by 𝑓(𝑥) = 3𝑥² + 1. Find the inverse function 𝑓−1(𝑥).

-π/4 π/4 π/2 3π/4 π

-1

1

2

3

4

5

x

y

• Functions and graphs

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  , 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

• Functions and graphs

Calderglen High School - Mathematics Department

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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) ∅ ∩ { 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 ⊂ A (c) {1} ⊂ A

(d) 0 ∈ ∅ (e) A ⊂ A (f) 1 ∈ A

• Functions and graphs

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 ∪ Q (c) P’

(d) Q’ (e) (P ∩ Q)’ (f) (P ∪ Q)’

(g) P ∩ Q’ (h) P’ ∩ Q (i) P ∩ ∅

O2 I have investigated domains and ranges.

1. State a suitable domain for the following functions:

(a) 𝑓(𝑥) = 𝑥2

𝑥−1 (b) 𝑓(𝑥) =

4𝑥 − 2

2𝑥 − 3 (c) 𝑓(𝑥) =

2𝑥 + 7

𝑥2− 16

(d) 𝑓(𝑥) = 𝑥2− 5𝑥 + 4

𝑥2+ 8𝑥+12 (e) 𝑓(𝑥) = √10 − 𝑥 (f) 𝑓(𝑥) = √𝑥2 + 3𝑥

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

(a) 𝑓(𝑥) = 3𝑥 − 4 ; 𝑥 ∈ { 2, 3, 4, 5 }

(b) 𝑓(𝑥) = 𝑥2 − 3𝑥 + 4 ; 𝑥 ∈ { −2, −1, 0, 1, 2 }

O3 I can determine a composite function.

1. Given 𝑓(𝑥) = 2𝑥 − 3, 𝑔(𝑥) = 𝑥² and ℎ(𝑥) = 𝑥² + 4, find the following

functions:

(a) 𝑓(𝑔(𝑥)) (b) 𝑔(𝑓(𝑥)) (c) ℎ(𝑓(𝑥))

(d) 𝑓(𝑓(𝑥)) (e) 𝑔(ℎ(𝑥)) (f) ℎ(ℎ(𝑥))

• Functions and graphs

Calderglen High School - Mathematics Department

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2. Given 𝑓(𝑥) = 𝑥 − 2, 𝑔(𝑥) = 2

𝑥2 and ℎ(𝑥) =

4

𝑥+1 , find the following

functions:

(a) ℎ(𝑓(𝑥)) (b) 𝑔(𝑓(𝑥)) (c) 𝑓(ℎ(𝑥))

(d) 𝑓(𝑔(𝑥)) (e) 𝑔(ℎ(𝑥)) (f) ℎ(ℎ(𝑥))

3. Given 𝑓(𝑥) = 𝑥 + 2, 𝑔(𝑥) = 𝑒𝑥 and ℎ(𝑥) = tan 𝑥, find the following

functions:

(a) 𝑔(𝑓(𝑥)) (b) 𝑔(𝑔(𝑥)) (c) ℎ(𝑓(𝑥))

4. Given 𝑓(𝑥) = 3𝑥2 + 2𝑥 − 1, 𝑔(𝑥) = sin 𝑥 and ℎ(𝑥) = log4 𝑥, find the

following functions:

(a) 𝑓(𝑔(𝑥)) (b) ℎ(𝑓(𝑥)) (c) 𝑔(𝑔(𝑥))

5. Two functions f and g, are defined by 𝑓(𝑥) = 2𝑥 + 3 and 𝑔(𝑥) = 2𝑥 – 3,

where x is a real number.

(a) Find expressions for 𝑓(𝑔(𝑥)) and 𝑔(𝑓(𝑥)).

(b) Determine the least possible value of the product 𝑓(𝑔(𝑥)) × 𝑔(𝑓(𝑥)).

6. Functions 𝑓(𝑥) = 3𝑥 − 1 and 𝑔(𝑥) = 𝑥² + 7, are defined on a set of real

numbers.

(a) Find h(x) where ℎ(𝑥)

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