Topic 4 Revision Notes
Functions
IB Math Studies
Topic 4.1 Domain, Range and Function Mapping
Functions are relationships which can be illustrated by mappings. Values go into the function and values come
out of the function. The entire set of numbers that can be entered are called the domain and the entire set of
numbers that output from the function is called the range.
Often it may be easier to graph a function to find the domain and range.
A mapping consist of two sets and a rule for assigning to each element in the first set one or more elements in
the second set.
Example
The mapping below is of the form f : x  x 2  1 and maps the elements of x to elements of y.
a) List the elements of the domain of f.
b) List the elements in the range of f.
c) Find p and q
Topic 4.2 Linear Functions
Linear functions always graph a line and are often written in the form y  mx  b where m is the slope (a.k.a.
gradient) of the function and b is the y-intercept.
A line with positive slope and a positive y-intercept
A line with positive slope and a negative y-intercept
A line with negative slope and a positive y-intercept
A line with negative slope and a negative y-intercept
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Topic 4 Revision Notes
Functions
IB Math Studies
Horizontal line


Vertical line
Horizontal lines are always in the form y = c,
where c is a constant.
The slope of a horizontal line is zero.


Vertical lines are always in the form x = c, where c
is a constant.
The slope of a vertical line is undefined.
Topic 4.3 Quadratic Functions
You need to be familiar with two different forms of a quadratic function:
Standard form
y  ax 2  bx  c
Vertex form
y  a ( x  h) 2  k
The graph of every quadratic is a PARABOLA (u-shaped)
vertex:

b
2a
,f
 2ab  
vertex: (h, k)
o
o
b
2a
  b   b 
, f
To find the vertex by hand: 
 
 2 a  2a  
The axis of symmetry: x 
1) The equation must be in the form
y  ax 2  bx  c
x-intercepts
(zeros, solutions)
2) The x-coordinate is equal to
b
2a
3) Plug the x-coordinate back into the
function, f(x), to get the y-coordinate of the
vertex.
o
o
o
To find the solutions (zeros/x-intercepts) by hand:
1) Set the equation equal to zero
2) Factor
3) Solve
 You will have two solutions
To find the solutions in the calculator:
1) Type the equation in Y=
2) Calculate
3) 2: Zero
To find the vertex in the calculator
1) Type the equation in Y=
2) Calculate
3) 3: Minimum (if the parabola opens up) or
4: Maximum (if the parabola opens down)
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Topic 4 Revision Notes
Functions
Diagram 1:
Diagram 2
Some observations about diagram 1.
 The y-intercept is -3 which is the same as the cvalue of the equation.
 The x-intercepts (also known as “zeros”) are at -1
and 3.
 Halfway between -1 and 3 is the x-coordinate of
the vertex; x = 1
 If you evaluate y(1) you will get the y-coordinate
of the vertex.
o y(1) = 12 – 2(1) – 3
o y(1) = -4
 if you set y = 0, then you can factor the equation
and solve for x:
o x2 – 2x – 3 = 0
o (x – 3)(x + 1) = 0
o x = 3 and x = -1
o These are the x-intercepts.
Some observations about diagram 2.
 The y-intercept is 6 which is the same as the c-value
of the equation.
 The x-intercepts (also known as “zeros”) are at 1
and 6.
 Halfway between -1 and 3 is the x-coordinate of the
vertex; x = 3.5
 If you evaluate y(3.5) you will get the y-coordinate
of the vertex.
o y(3.5) = (3.5)2 – 7(3.5) + 6
o y(3.5) = -6.25
 if you set y = 0, then you can factor the equation
and solve for x:
o x2 – 7x + 6 = 0
o (x – 6)(x – 1) = 0
o x = 6 and x = 1
o These are the x-intercepts.
Diagram 3
IB Math Studies
Some observations about diagram 3.
 Because the a value is negative, the graph is now opening down.
 The y-intercept is 10 which is the same as the c-value of the
equation.
 the x-intercepts (also known as “zeros”) are at -2 and 5.
 Halfway between -2 and 5 is the x-coordinate of the vertex; x = 1.5
 If you evaluate y(1.5) you will get the y-coordinate of the vertex.
o y(1.5) = -(1.5)2 + 3(1.5) + 10
o y(1.5) = 12.25
 if you set y = 0, then you can factor the equation and solve for x:
o -x2 + 3x + 10 = 0
o x2 – 3x – 10 = 0
o (x – 5)(x + 2) = 0
o x = 5 and x = -2
o These are the x-intercepts.
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Topic 4 Revision Notes
Functions
IB Math Studies
Guided example
The diagram shows the graph of y = x2 – 2x – 10
(a) Factor x2 – 2x – 10
(b) The point (0, a) is on the graph. Find the value of a.
(c) Find the coordinate of the minimum point of the graph.
Some quadratic equations don’t factor. Another way to solve them is to use the quadratic formula. This is given
to you on the formula sheet: x 
 b  b 2  4ac
2a
Topic 4.4 Exponential functions
Exponential functions are functions where the unknown value, x, is the exponent.
For the “mother function” the following is true:
 domain: all real numbers
 range: y > 0
 y-intercept: (0, 1)
 asymptote: y = 0
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Topic 4 Revision Notes
Functions
IB Math Studies
Exponential functions, like all functions, can be TRANSFORMED (translated, stretched, reflected).
Other examples of exponential functions:
An example of a negative exponential function:
y = 2-x – 2
y=2 –2
x
All exponential curves have asymptotes.
These are simply lines that the curve is heading towards but will never actually reach. In the two diagrams
below the asymptotes have been drawn on in dotted lines with the equation given below.
EXAMPLE 1: y = 2x – 1
Asymptote: y = -1
EXAMPLE 2: y = 2-x + 2
Asymptote: y = 2
Topic 4.5 Trigonometric functions
Sine and cosine curves are often used to model real life situations, such as hours of sunlight or tidal times. The
basic sine and cosine curves are shown below.
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Topic 4 Revision Notes
Functions
IB Math Studies
You need to be familiar with the graphs and properties of the sine and cosine functions in the following form:
f ( x)  a sin bx  c
f ( x)  a cos bx  c , where a, b, c, Q
Remember that the parameters affect the curve in the following ways:
a is the amplitude
b is the number of cycles between 0° & 360° and period 
360
b
c is the vertical translation.
Basic curves
y = sin x
y = cos x
Notice that for both f(x) = sinx and f(x) = cosx the DOMAIN is all real numbers and the RANGE is -1 ≤ y ≤ 1
Just as with any type of function, you can transform trigonometric functions by translating it, reflecting it or
stretching it.
Vertical translation
Adding a number to the function causes the curve to translate up.
Subtracting a number from the function causes the curve to translate down.
y = (sin x) + 3
y = (cos x) - 3
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Topic 4 Revision Notes
Functions
IB Math Studies
Vertical Stretch (changing the amplitude)
Multiplying the function by a number causes the curve to be stretched vertically; in other words, the amplitude has
changed. The amplitude is the distance between the principle axis of the function and a maximum (or a minimum).
y = 2sin x
y = 2cos x
Horizontal stretch (changing the period)
Multiplying x by a number causes the curve to be stretched horizontally.
y = sin (3x)
y = cos (2x)
Example
1 
x 3.
2 
Consider the function f ( x)  2sin 
a) Determine the amplitude
b) Determine the period
c) Sketch the graph of y  f ( x) on the grid below.
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Topic 4 Revision Notes
Functions
IB Math Studies
Topic 4.7 Sketching functions
Accurate graphing tips
o Use your calculator to help you. Be sure to use parenthesis where necessary.
o Label both the x and y axes (e.g. “time” and “height”)
o Include the scale on both axes. Sometimes you will be told what scale to use (e.g. “let 1 centimeter
represent 1 meter”). Be sure that you label this information.
o Use your calculator to assist you.
o Graph the function in your calculator; use the TABLE to get some points to plot. Remember that in
Table Setup (TBLSET) you can change the Indpnt to Ask and then you can choose which values
you want to plot.
Example
Consider the function f ( x) 
a)
b)
c)
d)
e)
2x
.
x 1
Find the y-intercept
Determine the minimum value of f(x) for x > 1.
Write down the equation of the vertical asymptote.
Calculate the value of f(5)
Sketch the graph of y  f ( x) for 4  x  7 showing
all the features found above.
Topic 4.8. Using the GDC to solve equations
Solving Functions with the GDC (TI-84)
*when you use the calculator to solve equations you will only get decimal approximated answers, not exact
answers.
 Type one side of the equation in y1
 Type the other side of the equation in y2
 Calculate > Intersect
Example.
Solve a ) x  2 
1
x
b) 5 x  3x
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