IB Math Studies
Topic: Functions
Chapters 16-18 of textbook
IB Course Guide description
Domain, Range and Function Mapping
• Domain: The set of values to be put
into a function.
– In other words the set of possible x
values
• Range: The set of values produced by a
function.
– In other words the set of possible y
values
Identify the domain and range of the
following functions
a.
b.
Check your answers
a. Domain: x ≥ -1
Range: y ≥ -3
b. Domain: any real number
Range: y ≤ 1
• A mapping diagram is a simple way to illustrate
how members of the domain are “mapped” onto
members of the range
– It shows what happens to certain numbers in the
domain under a certain function
This mapping for example
shows what happens to
the domain {-2, 0, 1, 2, -1}
under the function f(x) = x²
For a relationship to be a function, each member of the domain can
only map on to one member of the range; but it is ok for different
members of the domain to map onto the same member of the range
The mapping below is of the form and
maps the elements of x to elements of y.
• List the elements of the domain of f.
• List the elements in the range of f.
• Find p and q
Check your answers
• Domain: {q, -1, 0, 1, 3}
• Range: {5, 2, 1, p}
• q=2
• p=10
Linear Functions
• Always graph a line and are often written in the
form of y = mx + b
– Where m = slope or gradient
– Where b = y-intercept (the point where the line cuts
the y axis)
• ax + by = c is the rearrangement of this first form
Graphs of a linear function
A line with positive slope and a A line with positive slope and a
positive y-intercept negative y-intercept
A line with negative slope and a A line with negative slope and a
positive y-intercept negative y-intercept
Horizontal line Vertical line
- Horizontal lines are always - Vertical lines are always in
in the form y = c or y = k , the form x = c or x = k, where c
where c or k are the constant. or k are the constant.
- The slope of a horizontal line - The slope of a vertical line is
is zero undefined.
• Intersection of lines:
– The point where two lines can be worked out
algebraically by solving a pair of simultaneous
equations
• Finding the equation of a line
– You need to know its gradient and a point
• Can substitute into y = mx + b
• Use the formula y - y₁ = m(x - x₁) where (x₁ , y₁) is the
point
Example – equation of the line
– A line goes through (2,3) and (5,9) – what is its
equation
• Substitute into y = mx + b
– Gradient = = 2
– So y = 2x + b
– Substitute (2,3)
– 3 = 2(2) + b
– b = -1
– The equation is y = 2x - 1
• Use the formula y - y₁ = m(x - x₁) where (x₁ , y₁) is the point
– y - 3 = 2(x -2)
– y – 3 = 2x - 4
– y = 2x - 1
Quadratic Functions
• Two different forms
Standard form ² Vertex: (
Vertex form ² Vertex: (h,k)
• The graph of every quadratic function is a
parabola (u-shape)
X-intercepts:
(zeros,
solutions)
– To find the solutions (zeros/x-intercepts) by hand:
• Set the equation equal to zero
• Factor
• Solve
• You will have two solutions
– To find the solutions in the calculator:
• Type the equation in Y=
• Calculate
– 2: Zero
b
x
– The axis of symmetry: 2a
b b
• To find vertex by hand: , f
2a 2a
• The equation must be in the form ²
• The x-coordinate is equal to x b
2a
• Plug the x-coordinate back into the function, f(x), to get
the y-coordinate of the vertex.
– To find the vertex in the calculator
• Type the equation in Y=
• Calculate
• 3: Minimum (if the parabola opens up) or 4: Maximum
(if the parabola opens down)
• Example
• The y-intercept is -3 which is the same as
the c-value 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.
y(1) = 12 – 2(1) – 3
y(1) = -4
• If you set y = 0, then you can factor the
equation and solve for x:
x2 – 2x – 3 = 0
(x – 3)(x + 1) = 0
x = 3 and x = -1
These are the x-intercepts.
Quadratic formula
• Some quadratic equations do not factor
– To solve them use the quadratic formula
2
b b 4ac
x
2a
– This is given to use in the formula sheet the day of
the exam
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
Growth Decay
y = 2-x – 2
y = 2x – 2
The positive exponent The negative exponent
represents growth represents decay
• Exponential graphs are asymptotic
– They get closer and closer to a line but never reach it
EXAMPLE: y = 2x –
1
• If the equation is in the form y = ax
then the asymptote is the x-axis or 0
• If the equation is in the form y = ax + c
then the asymptote is the x = c
• The c in this equation shows the
movement upwards or
downwards of the graph
• In this example the -1 moved the
graph down one on the y axis
Asymptote: y = -1
Trigonometric Functions
Sine function Cosine function
a is the amplitude
c is the vertical translation
b is the number of cycles between 0° & 360° and period =
y = sin x y = cos x
• 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
• 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)
Sketching Functions
• Important tips
– Use your calculator to help you
• Set up the “window” correctly to see the part of the
graph that you need
• Remember parenthesis
– If you are not careful you could type an equation different
than the one the test is asking
– Label both x and y axes
– Include the scale on both axes
– Graph function in your calculator first
• Use the TABLE to get some point to plot
Using a GDC to solve equations
1. Type one side of the equation in Y1
2. Type the other side of the equation in Y2
3. Calculate – Intersect (option 5)
Remember the rules for sketching functions
