Download Functions and graphs booklet

STUDY KNOWHOW
PROGRAM
STUDY AND LEARNING CENTRE
Functions &
Graphs
Contents
Functions and Relations .............................................................................................................................................................. 1
Interval Notation ............................................................................................................................................................................ 3
Graphs: Linear Functions ............................................................................................................................................................ 5
Lines and Gradients ....................................................................................................................................................................... 7
Graphs: Quadratic Functions ..................................................................................................................................................... 9
Hybrid Functions ......................................................................................................................................................................... 11
Absolute Value Functions ........................................................................................................................................................ 13
Inverse Functions ........................................................................................................................................................................ 15
Graphs and Transformations ................................................................................................................................................. 17
FUNCTIONS AND RELATIONS
Relations
A relation is a set of ordered pairs.
(1, 2), (2, 6), (3, 4), (x, y), (2,4) are ordered pairs and {(1, 2), (2, 6)}, {(2, 6), (3, 4), (2,4)} and {(1, 2),
(2, 6), (3, 4), (x, y), (2,4) } are relations.
The domain of a relation is the set of first elements or the x-values of the ordered pairs.
For the relation {(1, 2), (2, 6), (3, 4), (x, y), (2,4)} domain Dx = {1, 2, 3, x}.
The range of a relation is the set of second elements or the y-values of the ordered pairs.
For the above ordered pairs the range Ry = {2, 6, 4, y}.
A rule that links the domain and range may be used to define the relation:
S  x, y: y  x , x  R 
The set of
ordered
pairs
the rule
joining the
domain and
range.
any restrictions
placed on the
domain or range.
This relation, called S, includes all ordered pairs that have y greater than x. (1,2), (1,100), and (⎯1,0)
are three of the ordered pairs in this relation.
The domain of S is Dx = R and the range of S will also be R.
Functions
A function is a special type of relation.
Each point in the domain of a function has a unique value in the range; each value of x may have only
one value of y.
The relation {(⎯1, 2), (⎯1, 4), (1, 6), (2, 8), (3, 10)} is not a function because there are two distinct
ordered pairs with x = ⎯1.
The relation {(⎯1,1), (0,2), (1,3), (2,5), (3,5)} is a function because each x value has only one
corresponding y value.
NB: If we have the graph of a relation it will be a function if any vertical line passes through only one
point.
Most functions are defined by a rule that links the x and y values, y is a function of x, or y = f(x).
The ordered pairs (0, 3), (2, 5), (⎯2, 1) and many
more belong to the function {(x,y): y= f(x) = x + 3}.
The graph of a function is way to show all the
ordered pairs that belong to the function.
1
Examples
1. State the domain and range: {(x,y): y = x2}. Is this relation a function?
y
20
Because x can be any real number and we can see
from the graph that it extends indefinitely to the
left and right: Dx = R
We can also see from the graph that the smallest
value of y will be 0 but the graph extends
indefinitely upwards: Ry {y:y≥0}
A vertical line drawn anywhere will pass through
only one point of the graph so the relation is a
function.
15
10
5
-4
2.
-2
4 x
2
The graph of x2 + y2 = 4 is shown. State the domain and range of this relation. Is the relation
a function?
y
4
Dx = {x: -2 𝑥
}
Ry: = {y: -2 𝑦
}
The relation is not a function because there
are many places where a vertical line
would pass through two points
eg: (0, 2) and (0, -2) both belong to the
relation.
2
-4
-2
4 x
2
-2
-4
Exercises
State the domain and range of the following relations, and whether each is also a function.
1. {(1, 0), (0,1)}
2. {(3, 2) (5, 5), (1, 5), (3, 9), (5, 10)}
3. {(x, y):
, x ≥ 0}
4. {(x, y): y = sin(x) }
y
y
2
1
1
0.5
2
4
6
8
10 x
-2
-1
2
4
6
x
-0.5
-2
-1
-3
5.
{(x, y): y = ex}
Answers
1. Dx = {0, 1} Ry = {0, 1}, function
3. Dx {x: x ≥ 0} Ry = {y: y 2}, function
5. Dx = R Ry = {y: y>0}, function
6. {(x, y): x2 + y2
9, x ≧ 0}
2. Dx = {1, 3, 5} Ry = {2, 5, 9, 10}, not a function
4. Dx = R Ry = {y: -1
}, function
6. Dx = {x: 0
} Ry = {y : ⎯3
}, not a function
2
INTERVAL NOTATION
Often the domain of a function will be restricted to a subset of R, with specific endpoints.
This subset is called an interval.
Closed intervals
If the end points are included it is called a closed interval . When representing a closed interval on a
real number line solid circles are used for the endpoints to indicate they are included:
a
x
b
and [ , ] are other ways to denote the same interval.
Open intervals
If the endpoints are not included, the interval is called an open interval. When representing an open
interval on a real number line open circles are used for the endpoints to indicate they are not included:
a
x
b
a < x < b and (a, b) are other ways to denote the same interval
Examples
Interval Notation
Inequality
Notation
Line Graph
x
[ , )
a
[ ,
)
b
x
≥
a
(
x
, )
b
NB:

In interval notation the smaller number is always written to the left i.e. [

The symbol
(infinity) is not a numeral.
is the concept of continuing indefinitely to the
right and
is the concept of continuing indefinitely to the left.
Hence we cannot write [ , ], [
, ] or
,
etc.
, )
( ,
Examples
1. Write the following in inequality notation and graph on a real number line.
(a) [ , )
Inequality notation
(b)
Graph
Graph
(
, ]
Inequality notation
x
-2
x
3
3
3
]
Multiple intervals
Two (or more) subsets of R, with end points a and b, and c and d, respectively, can also be represented
on a real number line.
Example
2
5
12
, ]
This is written in interval notation as (
].
( ,
Exercises
1. Write the following inequalities in interval notation and graph on a real number line.
(a)
0
(b)
(c)
2. Use interval notation and inequality notation to represent the following intervals.
(a)
(b)
x
x
-5
-3
0
3. Graph the following on a real number line and write in inequality notation.
(a) (
,
)
( ,
]
(b) [
, ]
[ ,9]
(c) (
, ]
( ,
)
Answers
1.
(a) [1, 10 ]
x
1
10
(b) [⎯ 6, ⎯ 4 )
x
-6
-4
(c) (5,  )
x
5
2. (a) (⎯, ⎯5 ], x  ⎯5
(b) (⎯3, 0 ), ⎯3 < x < 0
3. (a) ⎯ < x <⎯ 3 with 8 < x  13
x
-3
8
13
(b) ⎯ 1  x  4 with 6  x  9
-1
4
6
9
x
(c) - < x  3 with 6 < x < 
3
x
6
4
GRAPHS: LINEAR FUNCTIONS
Cartesian Plane





The Cartesian plane is defined by a pair of
mutually perpendicular coordinate axes or
number lines.
The horizontal axis is the x axis and the
vertical axis is the y axis.
Their point of intersection is called the origin.
Points are defined by coordinate pairs (x, y).
Coordinate pairs are ordered pairs: (3,1) is
not the same point as (1,3)
Linear Graphs
If the rule that generates a function can be arranged in the form y = mx + c then the function is a
linear function . The graph of a linear function is a straight line. Every point on the line represents an
ordered pair (x,y) that satisfy the rule for the function. Just two of these ordered pairs are needed to
graph the function.
The intercepts of a line are the points (0,y) and (x,0) where the line crosses the axes. These points are
useful for graphing a linear function.
The graph of the line represented by
y = – 2 is shown together with the
intercepts.
Examples
1.
Sketch the graph of the function {(x,y): y = 3x + 6}
We need to find any two points on the line
i.e. points that satisfy the rule y = 3x + 6.
The x- and y-intercepts are usually the easiest:
y-intercept: x = 0  y = 3(0) + 6 = 6
x-intercept: y = 0  0 = 3x + 6  x = ⎯2
The line passes through the points (0, 6) and (⎯2, 0)
5
2.
Sketch the graph of {(x, y): 3x – 4y = 0}
y-intercept: x= 0  3(0) – 4y = 0  ⎯4y = 0  y = 0.
The point (0,0) is on the line. But the y –intercept is also the x-intercept so a different
x –value is chosen to find a second point…
x = 2  3(2) – 4y = 0
 6 – 4y = 0
 4y = 6

y=
The point (2, ) is on the line.
Exercise
Sketch the graphs of the following linear functions.
(a) {(x,y): y = 4x +3}.
(b) {(x,y): 3y + 2x = 9}
(c) {(x,y): 2y + x = 0}
Answers
(a)
(b)
(c)
6
LINES AND GRADIENTS
The gradient of a line gives a measure of its slope, usually denoted m. This value may be positive:
or negative:
The gradient can be found:

directly from the graph (

by using the formula

by rearranging the equation of the line to the form y = mx +c , then m is the gradient.
The graph of the line y =
)
where ( ,
+ 2 is shown.
) and ( ,
) are two points on the line.
We can see
=
=
OR
The points (2,2) and (-2,0) are on the line:
𝑚
rise
𝑚
𝑦
𝑥
0
𝑦
𝑥
=
=
NB: It doesn’t matter which of the points are chosen as
(x1, y1) and (x2, y2)
run
𝑦
𝑥
OR
+2  𝑦
𝑥
and 𝑚
NB: when a line is in the form 𝑦
intercept
𝑚𝑥
𝑐, 𝑐 is the y-
Examples
1. Find the gradient of the line joining the two points (1,3) and (4,5).
=
=
2. Find the gradient of the line
Because the equation is in the form y = mx + c we can determine by inspection that
3.
Find the gradient of the line shown in the graph
The gradient of the line will be negative (
)
Drawing a rise and run triangle on the diagram enables us to see
7
=
⎯
=
⎯2
Parallel and perpendicular lines
Parallel lines have the same gradient. Perpendicular lines have gradients that are negative reciprocals.
y = ⎯3x – 1 and y = ⎯3x + 3 are parallel lines. y = ⎯ 3x – 1 and y =
m1 = m2
m1
⎯
or
+ 7 are perpendicular lines.
m1 × m2 = ⎯1
Horizontal lines have m = 0 and an equation of the form y = c.
Vertical lines have a gradient which is undefined and an equation of the form x = k.
Example
Draw the graph of the line that passes through (1,1) and is perpendicular to the line

.
Therefore the gradient of the new line is ⎯2.
Equation of new line is of the form y = ⎯2x + c.
Substitute (1,1) to find c: 1 = ⎯2(1) + c  c = 3
Equation of line is y = ⎯2x + 3
The line can be plotted using intercepts or by using
=
and the point (1,1).
Exercise
1. Find the equations of the lines shown in the graphs
(a)
(b)
2.
(c)
Graph the line that passes through the point (1,1) and is parallel to the
line y =  2x + 7.
Answers
1(a) y = 3
(b) y = x + 5
(c) y = ⎯0.5x + 2
2.
8
GRAPHS:
QUADRATIC FUNCTIONS
A quadratic function has the form f(x) = ax2 + bx + c, a  0. The graph of a quadratic function is
called a parabola.
Sketching a parabola
Find
 the orientation a < 0  , a > 0  .
 the turning point or vertex
 y-intercept [put x = 0]
 x-intercepts ( if any) [put y = 0, then solve the quadratic equation]
A parabola is symmetrical about a vertical line through the vertex.
Turning point
.
The x -coordinate of the vertex is given by x =
To find the y – co-ordinate substitute the value of the x- co-ordinate in the equation.
Example: Find the turning point of the parabola y = 3 – 2x – x2.
First rearrange to the form f(x) = ax2 + bx + c: y = – x2 – 2x + 3  a = ⎯1, b= ⎯2, c= 3
x=
=
= ⎯1 is the x-coordinate of the turning point
(⎯1)2
y = 3 – 2(⎯1) –
=4
The turning point is (⎯1, 4)
If the quadratic function can be rearranged to the form y = a(x ⎯ h)2 + k by completing the square
Then (h, k) is the turning point.
Eg: y = x2 – 6x + 13  y = (x – 3)2 + 4. The turning point of the parabola is (3,4)
Intercepts
The y intercept is the value of c in the equation y = ax2 + bx + c.
The x intercepts can be found by solving the equation ax2 + bx + c = 0
9
Examples
1. Sketch y = x2 + 3
a = 1, b = 0, c = 3
Orientation: a > 0
 the parabola opens upwards
Turning point: y = x2 + 3 = (x + 0)2 + 3
 (0, 3) is the turning point.
y-intercept: c = 3
x-intercepts: x2 + 3 = 0 has no solutions
so there are no x-intercepts
2. Sketch y = x2 +2 x  8
a = 1, b = 2, c = ⎯8
=
Turning point: x =
= ⎯1
X = ⎯1  y (⎯1)2
(⎯ ) –
⎯9
The turning point is (⎯1, ⎯9)
y-intercept: c = ⎯8
x-intercepts: x2 + 2x – 8 = 0
 (x + 4)(x – 2) = 0
 x ⎯ or x = 2
NB: Because of symmetry (⎯2, ⎯8) is also a point
on the graph
Examples
1. State the turning point of the following graphs:
(a)
(b)
(c)
2. Sketch the graphs of:
(a)
(b)
Answers
1. (a) (1, 5)
2.
(a)
(b) (0,3)
(
)
(
)
(d)
(c)
(d) (5,  3)
(c) (6,0)
(b)
(c)
10
(
)
HYBRID FUNCTIONS
Functions with a restricted domain
Functions which have different rules for each subset of the domain are called hybrid functions.
Sometimes they are referred to as piecewise defined functions.
Consider the following functions and their graphs noting the restricted domains:
y = ⎯x, x
⎯1
y = 1, ⎯
x
y
-4
y
x, x ≥
y
y
4
4
4
2
2
2
-2
2
4
x
-4
-2
2
4
x
-4
-2
2
-2
-2
-2
-4
-4
-4
Functions defined in pieces
These ‘pieces’ can be put together to form the hybrid function
,
y = f(x) = { ,
,
≥
and the graph of y = f(x):
y
4
2
-4
-2
2
x
4
-2
-4
Example
Draw a sketch graph of
y = f(x) =
{
,
0
≥0
,
y
4
2
-4
-2
2
4
x
-2
-4
11
4
x
Exercise
Draw a sketch graph of
,
0
1. f(x) = {
,
≥0
,
2. f(x) = {
0
0
,
,
3. f(x) = {0,
,
,
{ ,
,
4. f(x) =
Answers
1.
2.
y
-4
y
4
4
2
2
-2
2
4
x
-4
-2
2
-2
-2
-4
-4
3.
4
4.
y
-4
y
4
4
2
2
-2
2
4
x
-4
-2
2
-2
-2
-4
-4
12
4
x
x
ABSOLUTE VALUE FUNCTIONS
The absolute value of a number | | gives a measure of its size or magnitude regardless of whether it is
positive or negative. If a number is plotted on a number line then its absolute value can be considered
to be the distance from zero.
Examples
(i) | | = 2
(ii) | | = 2
| =| | =1
(iii) |
| |=8+1=9
(iv) | |
(v) | | = 7  x = 7 or x = ⎯7
Graph of the absolute value function
The absolute value function is a hybrid function defined as follows:
,
0
f:R →R where f(x) = | | = {
, ≧ 0
with graph
y
NB: The domain of f(x) = |𝑥| is R
and the range is R+ {0}
4
2
-4
-2
2
x
4
-2
-4
The graph of y = | | may be translated in the same way as the graphs of other functions.
Compare the graphs of the following functions with that of y =| |:
1. y = |
|
2. y = | | + 1
y = | | translated horizontally
2 units to the right
3. y = 3 ⎯ | | = ⎯| | + 3
y = | | translated vertically
one unit up
y = | | reflected in the x-axis
followed by a vertical shift of 3 units
To sketch the graph of y = | ( )| we need to sketch the graph of y = f(x) first and then reflect in the
x-axis the portion of the graph which is below the x-axis.
Example
Sketch { (x,y): y =|
|}
The graph of this function is the graph of y = x2 – 1 with the portion below the x-axis reflected across
the x-axis.
y
4
2
-4
-2
2
4
x
-2
-4
13
Equations and inequalities involving | ( )|
Because y = | ( )| is a hybrid function two cases must be considered when solving equations and
inequalities. For inequalities it is helpful to know that:
 | | < a  ⎯a < x < a
 | | > a  x > a or x < ⎯a
Examples
| =3
1. Solve |
|x
| = 3  x – 2 = 3 or ⎯ (x – 2) = 3
ie
x = 5 or x = ⎯1
With an absolute value expression on each side of the equation it is easier to square both sides:
Solve |
2.
| =|
|  (2x+1)2 = (x – 5)2
| =|
|
|
4x2 + 4x + 1 = x2 – 10x + 25
3x2 + 14x – 24 = 0
(3x – 4)(x + 6) = 0
x=
or x = ⎯6
NB: Care must be taken when multiplying or dividing by a negative to reverse an inequality.
3. Solve |
|
|<4
|<4
|
| < 12
 ⎯12 < 2 – x < 12
 ⎯14 < ⎯x < 10
 14 > x > ⎯10 or ⎯10 < x < 14
Exercise
1. Evaluate (a) |
|
(b) | 9
|
2. Sketch the graph of (a) y = |
3. Find for x
Answers
1. a) 11
2. (a)
R
b) 5
|
{ :| |
(a)
c) ⎯1
(b)
d) 9
|
(d) |
(b) y = |
}
(b) { : |
|
(c)
y
y
4
4
2
2
2
2
4
x
-4
-2
2
4
x
-4
-2
2
-2
-2
-2
-4
-4
-4
3. (a) {⎯6,6}
(b) {x: x
(c) y = |
| ≥ }
4
-2
|⎯ | |
e) 6
y
-4
(c) | | ⎯ |
1}
{x: x ≥ 5}
14
4
x
(e) |
0|
|
| |
INVERSE FUNCTIONS
Definition of an inverse function
If
(x) is the inverse function of a one-to-one function f (x) then
(x) is the set of ordered
pairs obtained by interchanging the first and second elements in each ordered pair.
So if (a,b)
f then (b,a)
f ⎯1 and if f (a) = b then f ⎯1(b) = a
The domain of f is the range of f ⎯1 and the range of f is the domain of f ⎯1.
For example, the function f :R →R defined by y = f (x) =
has an inverse function with rule
(x) = 2x + 1.
So (3,1) belongs to f and (1,3) belongs to f ⎯1 , and (⎯7,⎯4) belongs to f and (⎯4,⎯7) belongs to f ⎯1 .
Graph of an inverse function
The graphs of any one-to-one function f and its inverse f ⎯1 are symmetric about the line y = x.
Finding an inverse function for y = f(x)
To obtain the rule for an inverse function swap the x and y coordinates in f and rearrange to express y
in terms of x:
Example
Find the inverse function of f where f (x )= 2 – 3x
y=
x=
x–2=
⎯x + 2 =
2 – 3x
2 – 3y
⎯3y
3y
[swap x and y]
[rearrange to make ‘y’ the subject]
=y

(x) =
15
Exercise
Find the inverse of each of the following one-to-one functions:
1) y = x + 5
2) y = 4x
3) y =
4) y = √
, x≥½
Answers
1)
2)
(x) = x – 5
(x) =
3)
(x) =
4)
(x) =
, x0
16
GRAPHS AND
TRANSFORMATIONS
The known graphs of some simple functions and relations can be used to sketch related, but more
complicated functions.
Some graphs that would be useful to remember are:
y
y
x
y
y
y
x
x
y   x  a  x  b  x  c 
√
x
x
x-intercepts: ⎯a, ⎯b, c
17
x
y
y
x
x
1
log
Circle of radius
y
y
1
x
ya
x
y
x
1
x2
To sketch a graph look for:
x and y intercepts
turning points
behaviour as x tends to
asymptotes (e.g. when the denominator of a fraction = 0)
Graphs should be named, axes labelled and any intercepts, turning points or asymptotes marked.
Reflections
( ) then
If
(
) is the reflection of the graph of about the y-axis
( ) is the reflection of the graph of about the x-axis
Examples
( )
( )
y
𝑦
10
𝑥
5
-4
-2
2
4
x
-5
𝑦
𝑥
-10
18
( )
3.
(
)
(
If ( )
, (
about the y-axis.
)
)
. The graph of
𝑦
is the graph of
𝑥
𝑦
𝑥
Translations
A translation may be a horizontal shift or a vertical shift.
Horizontal
The graph of
The graph of
(
(
Vertical
( )
( )
The graph of
The graph of
) is a shift of the graph
) is a shift of the graph
( )
( )
units to the right.
units to the left
is a shift of the graph
is a shift of the graph
( )
( )
units up.
units down.
Examples
log(
)
Basic graph is
y
log
Replacing x with (x 1) shifts the graph 1
unit to the right
4
y  log( x  1)
2
2
4
6
8
x
shifted 1 unit right
-2
19
reflected
(
)
Basic graph is
y
10
Replacing x with (x + 3) shifts the graph 3
units to the left
5
-6
-4
-2
(
x
2
-5
)
shifted 3 units left
Asymptote is at x = ⎯ 3, when x + 3 = 0
-10
Basic graph is
y
15
Adding 2 shifts the graph up 2
10
y  x2  2
5
shifted 2 units up
-4
-2
2
x
4
-5
Dilations
A dilation is a stretching or a squashing.
The graph of
( ) is a dilation of the graph of
axis.
The graph of
( ) is a dilation of the graph of
axis.
( ) by a factor of
( ) by a factor of
units parallel to the yunits parallel to the x-
Examples
Stretched by a factor of
2 parallel to the y -axis
y
4
y  2 x2
3
2
y  x2
1
-2
-1
1
2
x
-1
20
Each value of 𝑦
𝑥
is obtained by
multiplying the
corresponding value of
𝑦 𝑥 by 2.
e.g. x = 1 gives y = 1
for
𝑦 𝑥 and y = 2 for
𝑦
𝑥
(
)
(
)
(
Dilated by a factor of 1/2
parallel to the x -axis
)
In this case x values that
differ by a factor ½ give
the same y – values
(
e.g The value of y = 4 is
given by x = 3 for
) and by
𝑦 (𝑥
)
𝑥
for 𝑦 ( 𝑥
)
Combinations
More complicated graphs can be sketched by using combinations of dilation, reflection and
translation.
Dilations must always be considered before reflection and translation.
Examples
Basic graph is
y
10
Multiplying by 2 stretches the graph by a
factor of 2 parallel to the y axis
8
Adding 4 shifts the graph up 4 units
6
4
2
-2
-1
1
2
3
Stretched
by factor 2
vertically
x
-2
(
)
Basic graph is
y
-4
shifted 4
units up
-2
2
-2
-4
4
6
x
Replacing x with (x + 2) shifts the graph
2 units to the left.
Adding ⎯1 shifts the graph 1 unit down
The negative sign before the bracket
reflects the graph about the x – axis
(
-6
-8
reflected
about x
– axis
21
)
Shifted 2
units left
shifted 1
unit
down
√
Basic graph is
y
√
Replacing x with  x reflects the graph
about the y – axis.
6
Adding +2 shifts the graph up 2 units.
4
√
2
-8
-6
-4
x
-2
Shifted 2
units up
reflected
about
y – axis
-2
Exercises
Sketch the following graphs.
1.
(
4. (
2.
)
)
(
)
5.
)
log(
(
3.
)
6.
Answers
4
y
10
2
5
-4
-4
-2
2
4
4
-10
2
-2
x
-4
-5
1
-2
2
3
5
6
22
-6
4
6
x