FUNCTIONS AND THEIR
GRAPHS
Mrs. Aldous, Mr. Beetz & Mr. Thauvette
IB DP SL Mathematics
You should be able to…
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Use the features of your GDC to graph a variety of
functions
Use your GDC to investigate key features of
graphs, including maximum values, minimum values,
points of intersections, and zeros
Find a value of a function
Find the range of a functions on a given domain
Identify horizontal and vertical asymptotes in
functions where appropriate
Solve equations using a graphical approach
You should know…
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A function is usually denoted by a small letter of the
alphabet, such as f or g; the notation f(x) or g(x) is
the value of the function at x
The domain of a function is the set of x-values for
which a function is defined
The range of a function is best found by looking at
the maximum and minimum values of the graph of
the function on a given domain
Basic Functions
Substituting numbers into functions
A function can be written as:
f(x) = 4x - 3
Try some of these:
Substituting is replacing the x
so that,
1.f (x) =
f(5) = 4 ´ 5 - 3 = 17
Check these mentally:
f (3) = 4x3-3=9
f (0) = 4x0-3=-3
x+5
3
a) f(7)= 4
2.g(x) = x 2 - 3x + 2
a) g(3)= 2
b) g(-1)= 6
3.h(x) = 2(x + 3)2 - 5
a) h(1)= 27
f (-2) = 4x-2-3=-11
b) f(-2)= 1
b) h(-5)= 3
Solving functions
Solving a function involves finding Try some of these:
a solution for x.
f(x) = 4x - 3
Solve f(x)=0
This is a simple algebra problem:
4x - 3 = 0
4x = 3
x=
3
4
Check f(x)=17
x=5
1.f (x) =
f(x)=4
x+5
3
x=7
2.g(x) = x 2 - 3x + 2
g(x)=0
x=1 and x=2
3.h(x) = 2(x + 3)2 - 5
h(x)=4
x=-0.87 and
x=-5.121
Function vs. Mapping Notation
Drawing a graph and adjusting the window
Enter: Y1 = x – 3
Now take a look at the graph in “standard” view:
To pick another window to view the graph:
Now enter your own values and see how it
affects the viewing window when you look at
the graph.
Solving Points on a Graph
Using the graph: Y1 = x – 3
We can solve the following using the
CALC menu on the GDC:
More solving points on a graph
Now for a quadratic:
f (x) = x 2 - 3x - 28
1. Enter this quadratic and get a
window so that you can see all
the information needed.
2. Find both roots using CALC >
2:zero. You will need to give a
left and right bound and a guess.
x=-4 and x=7
3. Find the coordinates of the
minimum using CALC >
3:minimum.
(1.5,-30.25)
4. Find the y-intercept using
CALC > 1:value.
y=-28
5. Find f(4) using algebra or CALC
> 1:value.
y=-24
6. Find f(x)=5 using algebra or
CALC > 5:intersect.
x=-4.44
Graphs of Functions
The x-intercepts of a function are the values of x for which y = 0.
They are the zeros (i.e., solutions, roots) of the function.
The y-intercept of a function is the value of y when x = 0.
Graphs of Functions
An asymptote is a line that the graph approaches or begins to look
like as it tends to infinity in a particular direction.
vertical asymptote
x=2
y=2
horizontal asymptote
Graphs of Functions
To find vertical asymptotes, look for values of x for which the
function is undefined:
• If y =
f (x)
find where g(x) = 0
g(x)
• If y = loga ( f (x)) find where f (x) = 0
To find horizontal asymptotes, consider the behaviour as x ®±¥
Asymptotes
An asymptote is a line that a
graph gets closer to but never
reaches.
What is not available
for the domain?
x=2
Graph x=2.
1
- 3.
2- x
Use -2 £ x £ 4 and -10 £ y £ 10.
Sketch the graph of f (x) =
What number is not
in the range?
y=-3
1
f (x) =
-3
2- x
This part of the
functions is never 0,
so y=-3 is never
possible.
The asymptotes are
x=2 and y=-3.
Now try one on your own
1
1
+ .
2x - 3 2
Choose an appropriate window for the graph.
On your GDC graph of f (x) =
1. Use your GDC to find the root.
x=
1
2
2. Use your GDC to find the yintercept.
1
y = (0.166)
6
3. Use your graph or algebra to
find the two asymptotes of the
function.
x=
3
2
y=
1
2
Example
Let g(x) = x - 2x + x - 2.
4
3
2
(a) Write down the values of x for which g(x) = 0.
Example
Example continued…
Example continued…
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d
Domain and range
The domain and range will be examined further in later lessons.
For the purpose of this slide you may assume that:
Domain is the x-values of a function or graph (input).
Range is the y-values of a function or graph (output).
Domain and range
Domain
y+5
2
Range
5
5
5
8
11
1
0
-5
7
-3
-11
f (x) =2x - 5
1
x=
-3
This is a one-to-one relationship,
and can be a function with an
inverse.
f (x) = ( x - 3) + 4 x = y - 4 + 3
2
Domain
Range
8
20
-1
This is a many-to-one
relationship, and is a function
but without an inverse.
Domain and range continued
1
f (x) = - 2
x
Domain
x=
1
y+2
Range
1
-1
0
undefined
3
f (x) = x + 3
Domain
Range
-2
1
-1
5
3
6
-3
13
-1
x = y -3
2
3
-3
4
-4
A one - to - one function,
with an inverse if the
domain: x any number, x ¹ 0.
This is a one-to-many
relationship, and therefore is
not a function.
Domain and range continued
f (x) = x 2 - 4x + 5
Domain
x = y -1 + 2 A one-to-one relationship
is a function with an
Range
inverse.
1
3
2
-1
5
10
This is a many-to-one
relationship, and is a function
but without an inverse.
A one-to-many
relationship is a not a
function.
A many-to-one
relationship is a function
without an inverse.
Function - a rule that links
each member of the
domain to exactly one
member of the range.
Domain and range finder
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Cut out the graphs and
the domain and range
finder.
For the domain, we
want to look at the x
coordinates. Use the left
and right flaps.
For the range, we want
to look at the y
coordinates. Use the top
and bottom flaps.
Domain and range finder
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Use the flaps to help
you see what domain
and range the graph
has.
Domain and range finder
Function
Domain
Range
y = log x
{x | x > 0}
{y | y Î }
y = x2
{x | x Î }
{y | y Î }
y= x
{x | x ³ 0}
{y | y ³ 0}
x 2 + y2 = 16
{x | -4 £ x £ 4}
{y | -4 £ y £ 4}
y = ex - 5
{x | x Î }
{y | y > -5}
4
x +1
y =| x -1| -5
{x | x ¹ -1}
{y | y ¹ 0}
{x | x Î }
{y | y Î }
{x | -7 < x £ 7}
{y |1 < y £ 7}
{x | x Î }
{y | y Î }
y=
y=
1
x + 4, - 7 < x £ 7
2
y = x 3 - 2x 2 + x - 4
Restricting the domain
f (x) = ( x + 3) - 4
2
This is a function without
an inverse, as can be see
from the graph.
However by stating the
domain the function
becomes one-to-one and
has an inverse.
f (x) = ( x + 3) - 4, x ³ -3
2
Restricting the range
f (x) = x - 3
This is not a function as it
is a one-to-many
relationship as can be seen
from the graph.
However, by restricting the
range:
f (x) = x - 3, such that
f(x) is positive.
Domain & Range
Set vs. Interval Notation
Set vs. Interval Notation
Set vs. Interval Notation
Set vs. Interval Notation
You should know…
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The line x = a is a vertical asymptote if f(a) is
undefined and f (x) ® ±¥ as x ® a
Horizontal asymptotes tell us how the graph of a
function behaves when x gets very large positively
or negatively. They are often found in rational and
exponential functions.
Be prepared…
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Know the major features of your GDC.
When asked to sketch a graph, make effective use
of the zoom and trace features to ensure that you
are looking at an accurate representation of the
graph. Set your window equal to the given domain
When finding numerical values such as maximum
and minimum points, use the appropriate menus and
not the trace features of your GDC.
Information about sketching graphs
You may be asked to sketch a graph. This is usually after you have used a
GDC, but not always.
A sketch does not need to be accurate, but it does need to be neat and it
must show the vital information. You will not need graph paper.
Your sketch should be:
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about a third of a page in size,
drawn in pencil with a ruler for all the straight lines including the axes,
clearly labeled, this includes the axes.
On your sketch you must mark the following points:
•
where the graph intercepts the axes,
•
draw and label the asymptotes.
Sketching graphs
Sketch the following graphs:
1. f (x) = x 2 - 7x + 6
2. f (x) = x 3 + x 2 - 2x + 1
1
3. f (x) = 1+
x -2
4. f (x) = 2 x - 4
•
about a third of a page in size,
•
drawn in pencil with a ruler for all the straight
lines including the axes,
clearly labeled, this includes the axes.
•
On your sketch you must mark the following points:
1
5. f (x) = + 1
x
•
where the graph intercepts the axes,
•
draw and label the asymptotes.