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Graphs & Functions

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Higher Maths

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Graphs & Functions Higher

Graphs & Functons

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The diagram shows the graph of a function f. f has a minimum turning point at (0, -3) and a point of inflexion at (-4, 2).

a) sketch the graph of y = f(-x).

b) On the same diagram, sketch the graph of y = 2f(-x)


a) Reflect across the y axis

b) Now scale by 2 in the y direction-1 3 4

2

y = f(-x)

-3

y

x

4

y = 2f(-x)

-6

Hint



The diagram shows a sketch of part ofthe graph of a trigonometric function

whose equation is of the form

Determine the values of a, b and c

sin( )y a bx c

a is the amplitude: a = 4

b is the number of waves in 2 b = 2

c is where the wave is centred vertically c = 1

2a

1 in 2 in 2

1

Hint



Functions and are defined on suitable domains.

a) Find an expression for h(x) where h(x) = f(g(x)).

b) Write down any restrictions on the domain of h.

1( )

4f x

x

( ) 2 3g x x

( ( )) (2 3)f g x f x a)1

2 3 4x

1

( )2 1

h xx

b) 2 1 0x 1

2x

(2, 1)

(2, -1)

(2, 1)

5

y=f(x)

y= -f(x)

y= 10 - f(x)

Hint



a) Express in the form

b) On the same diagram sketch

i) the graph of

ii) the graph of

c) Find the range of values of x for

which is positive

2( ) 4 5f x x x 2( )x a b

( )y f x10 ( )y f x

10 ( )f x

a) 2( 2) 4 5x 2( 2) 4 5x 2( 2) 1x

b)

c) Solve:210 ( 2) 1 0x

2( 2) 9x ( 2) 3x 1 or 5x

10 - f(x) is positive for -1 < x < 5

Hint



The graph of a function f intersects the x-axis at (–a, 0)and (e, 0) as shown.

There is a point of inflexion at (0, b) and a maximum turningpoint at (c, d).

Sketch the graph of the derived function f

m is + m is + m is -

f(x)

Hint



Functions f and g are defined on suitable domains by

and

a) Find expressions for:

i)

ii)

b) Solve

( ) sin( )f x x ( ) 2g x x

( ( ))f g x

( ( ))g f x

2 ( ( )) ( ( )) 0 360f g x g f x for x

( ( )) (2 )f g x f xa) sin 2x ( ( )) (sin )g f x g x 2sin x

b) 2sin 2 2sinx x sin 2 sin 0x x

2sin cos sin 0x x x sin (2cos 1) 0x x 1

or2

sin 0 cosx x 0 , 180 , 360x 60 , 300x

Hint



The diagram shows the graphs of two quadraticfunctions

Both graphs have a minimum turning point at (3, 2).

Sketch the graph of

and on the same diagram

sketch the graph of

and( ) ( )y f x y g x

( )y f x

( )y g x

y=g(x)

y=f(x)

Hint



Functions are defined on a suitable set of real numbers.a) Find expressions for

i) ii)

b) i) Show that

ii) Find a similar expression for

and hence solve the equation

4and( ) sin , ( ) cos ( )f x x g x x h x x

( ( ))f h x ( ( ))g h x1 1

2 2( ( )) sin cosf h x x x

( ( ))g h x

for( ( )) ( ( )) 1 0 2f h x g h x x

4( ( )) ( )f h x f x a) 4

sin( )x 4

( ( )) cos( )g h x x

sin cos4 4 4

sin( ) sin cos xx x b) Now use exact values

Repeat for ii)

equation reduces to2

sin 12

x 2 1sin

2 2x

3,

4 4x

Hint



A sketch of the graph of y = f(x) where is shown.The graph has a maximum at A and a minimum at B(3, 0)

a) Find the co-ordinates of the turning point at A.

b) Hence, sketch the graph of Indicate the co-ordinates of the turning points. There is no need to calculate the co-ordinates of the points of intersection with the axes.

c) Write down the range of values of k for which g(x) = k has 3 real roots.

3 2( ) 6 9f x x x x

( ) ( 2) 4g x f x

a) Differentiate2( ) 3 12 9f x x x for SP, f(x) = 0 1 3x or x

when x = 1 4y t.p. at A is: (1, 4)

b) Graph is moved 2 units to the left, and 4 units up(3, 0) (1, 4)

(1, 4) ( 1, 8)

t.p.’s are:

c) For 3 real roots, line y = k has to cut graph at 3 points

from the graph, k 4

Hint



3( ) 3 ( ) , 0x

f x x and g x x

a) Find

b) If find in its simplest form.

( ) where ( ) ( ( ))p x p x f g x3

3( ) , 3

xq x x

( ( ))p q x

3( ) ( ( ))x

p x f g x f

a)3

3x

3 3x

x

3( 1)x

x

b)

33 1

3333

3

( ( )) x

xx

p q x p

9 3

33 3x x

9 3(3 ) 3

3 3

x x

x

3 3

3 3

x x

x

x

Hint



Part of the graph of is shown in the diagram.

On separate diagrams sketch the graph of

a) b)

Indicate on each graph the images of O, A, B, C, and D.

( )y f x

( 1)y f x 2 ( )y f x

a)

b)

graph moves to the left 1 unit

graph is reflected in the x axis

graph is then scaled 2 units in the y direction

Hint



Functions f and g are defined on the set of real numbersby

a) Find formulae for

i) ii)

b) The function h is defined by

Show that and sketch the graph of h. c) Find the area enclosed between this graph and the x-axis.

2( ) 1 and ( )f x x g x x

( ( ))f g x ( ( ))g f x

( ) ( ( )) ( ( ))h x f g x g f x 2( ) 2 2h x x x

a)

b)

2 2( ( )) ( ) 1f g x f x x 2( ( )) ( 1) 1g f x g x x

22( ) 1 1h x x x 2 2( ) 1 2 1h x x x x 22 2x x

c) Graph cuts x axis at 0 and 1 Now evaluate1 2

02 2x x dx

2unit

1

3Area

Hint



The functions f and g are defined on a suitable domain by

a) Find an expression for

b) Factorise

2 2( ) 1 and ( ) 2f x x g x x ( ( ))f g x

( ( ))f g x

a) 22 2( ( )) ( 2) 2 1f g x f x x

2 22 1 2 1x x Difference of 2 squares

Simplify 2 23 1x x

b)

You have completed all 13 questions in this section

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30° 45° 60°

sin

cos

tan 1

6

4

3

1

2

1

23

2

3

2

1

21

21

3 3

Table of exact values

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