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Straight Line
n5 straight line revision
At National 5, there are a few skills required with straight lines, however, at Higher
these skills are expanded upon.
At National 5 there were three key formulae for straight lines, these were:
๐ฆ
2
โ๐ฆ
1
๐ฅ
2 โ
๐ฅ
1
- The Equation of a Straight Line - ๐ฆ = ๐๐ฅ + ๐
- The Equation with Unknown Y-Intercept - ๐ฆ โ ๐ = ๐
At Higher, these formulae are the basis of complex problem-solving questions and
act as an introduction to new formulae.
Basic new formulae
At Higher, the first new formulae/ skills required are those of finding the distance
between two points on a straight line and finding the midpoint of a straight line.
These are both basic skills which are not assessed on their own but in the form of
larger more complex questions.
To find the distance, the Distance Formula is used, by subtracting the two xโs from
each other and the two yโs from each other, and then adding the two answers
together before square rooting that:
2
1
2
1
To find the midpoint of a straight line you add both x coordinates together and divide
them by 2 and do the same for the y coordinates. There is no formula for this skill as it
is only worth one mark.
Gradients & m=TANฮธ
At National 5 there were two gradient slopes, positive and negative, both of which gave
a numerical gradient and followed the equation ๐ฆ = ๐๐ฅ + ๐ ,however at Higher, there
are two other types of gradients a vertical and horizontal.
A vertical gradient runs parallel to the y axis and provides an undefined gradient (also
shown as 0 on denominator) as the slope is so minute, it is undefinable. Due to this a
vertical straight line does not follow the usual equation but instead follows: ๐ฆ = ๐ฅ.
A horizontal gradient runs parallel to the x axis and provides a zero gradient (also
shown as 0 on the numerator) as the line is running parallel to the axis and has no slope.
Due to this the line follows an equation of ๐ฅ = ๐ฆ.
The first higher formula for rectilinear shapes is, ๐ = ๐ก๐๐๐ this formula is used when
the gradient of a straight line creates an angle at the positive direction the x axis.
To apply ๐ = ๐ก๐๐๐ and calculate the angle at the positive direction of the x axis, there
are a series of steps that must be taken:
- Step 1: Use the gradient formula (from N5) to find the gradient
- Step 2: Substitute the gradient into the formula ๐ = ๐ก๐๐๐
- Step 3: Rearrange the formula to have the theta on the left and gradient on the
right.
- Step 4: Using inverse tan function find tan inverse of the gradient (this can be
non-calculator as well)
If the gradient is unknown but the angle is known the formula can be rearranged to
find ๐.
parallel lines
A perpendicular bisector is a straight line that cuts a given line at a right angle and at
its midpoint. To calculate the equation of the perp bisector, the gradient of the line
the bisector is intersecting must be found and turned into a perpendicular gradient.
Then, by inputting the perpendicular gradient and the coordinates of the midpoint
into the formula ๐ฆ โ ๐ = ๐
, the equation of the perpendicular bisector can be
found.
Point of Intersection (POI) is when two perpendicular lines intersect. To calculate the
POI, the equation of the two straight lines can be rearranged into simultaneous
equation and solved to give the coordinates of the POI.
COLLINEARITY
Three or more coordinates are said to be collinear if they sit on the one parallel
straight line.
To prove something is collinear; it must be shown that they have the same gradient
and share a common coordinate. This is done by calculating the gradient of point A
to point B and then the gradient of point B to point C. Then, if the gradients are the
same, a statement can be wrote saying: โSince gradient AB and BC were equal, the
lines are parallel and as they share the common point B, are collinear.โ
Recurrence Relations
RECURRENCE RELATIONS
A recurrence relation is a sequence that gives you a connection between two
consecutive terms. This connection can be used to find other subsequent terms.
Each term in the recurrence relation is expressed as the letter โU,โ followed by the
number of the term it is in the sequence, for example the fourth term in the relation
would be ๐
4
. The initial value in the relation is called ๐
0
A basic recurrence relation is expressed as: ๐ข
๐+ 1
๐
. In this ๐ข
๐
is the initial term
and ๐ is the constant. This means, to calculate a basic recurrence relation the ๐ would
be multiplied with the ๐ข
๐
to give the next term in the sequence.
A linear recurrence relation is used to calculate multiple terms in a sequence and
uses the previous term to calculate the next term. A linear recurrence relation takes
the form: ๐ข ๐+ 1
๐
- ๐. To calculate the terms, the constant must be multiplied by
the ๐ข 0
and then, the ๐ must be added on. The answer to this will be input back into the
form ๐ข ๐+ 1
๐
- ๐, with the answer that was calculated replacing the ๐ข
0
. This will
allow the next term to be found. This process will be repeated until the term the
question has asked for is found.
LIMITS
A recurrence relation can either be divergent or convergent. A divergent relation is a
recurrence relation that continues to decrease or increase for infinity. A convergent
relation is a recurrence relation that settles at a particular value after a period, the
Circles
EQUATION OF A CIRCLE
The equation of a circle can be expressed in one of two ways:
- Equation of a circle in bracketed form.
- Equation of a circle in expanded form.
The equation of a circle in bracket form is expressed as: (๐ฅ โ ๐)
2
2
2
The equation of a circle in expanded form is expressed as: ๐ฅ
2
2
To find the centre point of the circle from the equation in brackets, the ๐ value and
the ๐ value can be removed from the bracket and put into the opposite charge to
give the coordinates of the centre. To find the radius it can be squared.
To find the centre point of the circle from the expanded equation, ๐๐ฅ and ๐๐ฆ values
can be halved and put into the opposite charge to give the coordinates. To find the
radius the formula ๐ = โ๐
2
2
โ ๐ can be used.
CONCENTRIC CIRCLES
Concentric circles are circles which have the same centre but different radius. A point
within a concentric circle can lie inside, outside, or on the circumference of the circle.
To prove where a point lies there are three statements that can be used:
2
2
2
this statement means the point is inside.
2
2
2
this statement means the point is outside.
2
2
2
this statement means the point is on the circumference.
The way to prove where a point lies by following these steps:
2
and ๐ฆ
2
values together to give what the left-hand side of the
equation of the circle equals and then state whether it equals the right-hand
side of the equation, followed by the correlating statement which indicates
where the point lies.
INTERSECTION OF CIRCLES
Circles which are not concentric are called eccentric circles and can intersect with one
and other in one of three ways:
- No intersection.
- Intersection at one point.
- Intersection at two points.
To calculate how many times a circle intersects, find the radius of both circles and
calculate the distance between both the circles using the distance formula on the
centres of both circles. Then add both the radii together. If the answer is greater than
the distance, the circles do not intersect. If the answer is less than the distance, the
circles intersect twice. If the answer is equal to the distance, the circles intersect once.
Circles that intersect once can touch internally or externally.
INTERSECTION OF A CIRCLE AND A LINE
A straight line and a circle can intersect at either one point, two points, or no points at
all.
Differentiation
PREPARING TO DIFFERENTIATE
Differentiation is used in maths for calculating the rate of change.
The derivative (differentiated answer) uses the communication:๐
โฒ
(๐ฅ) or
๐๐ฆ
๐
A differentiation question can be asked in various ways:
โฒ
๐๐ฆ
๐
- Calculate the rate of change of...
- Find the derivative of...
- Calculate the gradient of the tangent to the curve...
To differentiate there are certain steps that must be taken to prepare for the
differentiation:
- Ensure there are no root signs.
- Ensure there are no brackets
- Ensure there are no values on the bottom of a fraction.
Once these steps are taken, differentiation can be done.
To differentiate the ๐ฅ must be multiplied by the power and one must be subtracted
from the power to give a new power. When differentiating number without
๐ฅ disappear and ๐ฅ values with a power of one become number without.
Basic differentiation can be applied into various questions but to do so any negative
powers and fractional powers must be removed from the derivative again.
RATE OF CHANGE
The derivative can be used to find the rate of change of one quantity compared to
another at one point.
To find the rate of change, the derivative must be found, and the value given for ๐ฅ
should be subbed into the derivative.
EQUATION OF A TANGENT TO A CURVE
In maths the equation of a tangent to a curve can be found using differentiation as the
derivative and the gradient are said to be equal.
To find the equation of the tangent to the curve two things must be found: the
gradient and the point. Sometimes the point will be given and other times only the
๐ฅ coordinate will be given, and the ๐ฆ coordinate will have to be found.
To find the gradient the equation of the curve must be differentiated and the point at
which the tangent touches the curve must be subbed in.
To find the point, the ๐ฅ coordinate at which the tangent touches the curve must be
subbed into the equation of the curve. This will give the ๐ฆ coordinate and hence the
point of contact.
Then to find the equation of the tangent to the curve the gradient and point can be
subbed into the straight-line formula: ๐ฆ โ ๐ = ๐(๐ฅ โ).
STATIONARY POINTS
Optimisation is the real-life application of differentiation and is used to find the
maximum and minimum values required.
Optimisation questions are split into a part a and a part b. Part a will ask for it to be
proven (using the diagram given) that the function of that shape is what they have
said it is. Part b will ask for the minimum or maximum amount of material required in
link with the scenario.
To answer Part b of optimisation, the procedure used to answer a stationary points
question is used but when factorising to find the ๐ฅ value, any negative value is
disregarded as negative values cannot be used.
INCREASING/DECREASING FUNCTIONS
When the curve representing the function is drawn it can be increasing or decreasing
at certain points. For any curve if
๐๐ฆ
๐๐ฅ
0 the function is increasing,
๐ฆ
๐๐ฅ
< 0 the function
is decreasing.
To calculate whether a function is increasing or decreasing, the function must be
differentiated, and the ๐ฅ coordinate given must be subbed into the derivative. If the
answer comes out greater than 0 the function is increasing at that point in the graph.
If the answer comes out less than 0 the function is decreasing at that point in the
graph.
DERIVED GRAPHS
Derived graphs are when the derivative is used to sketch a graph from its function.
When sketching a derived graph, the same procedure as used for stationary point
questions is used to find out the shape of the curve. Then to find out where the curve
cuts the ๐ฆ axis, sub ๐ฅ = 0 into the original equation of the curve and to find out where
the curve cuts the ๐ฅ axis, sub ๐ฆ = 0 into the original equation of the curve. Finally, all
this information is used to sketch a derived graph.
opened, and the first limit needs to be subbed into every ๐ฅ value in the integral. Then
place a subtraction sign and open another bracket. In the second bracket sub the
second limit into every ๐ฅ value in the integral. Finally, the brackets should be
simplified down and subtracted to reveal the answer.
AREA ABOVE & BELOW THE X AXIS
To calculate the area above and below the ๐ฅ axis, the definite integral of the two limits
above the ๐ฅ axis needs to be calculated and then the same for the two limits below
the ๐ฅ axis. The answer is then the answer for area above and area below added
together.
AREA BETWEEN TWO CURVES
To calculate the area between two curves there is a formulaic
structure: โซ
๐๐๐๐๐ ๐ฟ๐๐๐๐ก
๐ฟ๐๐ค๐๐ ๐ฟ๐๐๐๐ก
[
]
What this means is that the equation of the top curve and bottom curve must be
integrated as a definite integral. To do this the equations of both curves must be put
into separate brackets and multiplied out by the minus sign. This will give a new
equation to be integrated. This must then be integrated like a definite integral to give
the area between both the curves.
FINDING LIMITS
Sometimes the limits are not given and must be calculated. To do this the equations
of the curves must be set equal to each other and set equal to zero to get a quadratic
equation. This quadratic equation must then be factorised to give the limits.
DIFFERENTIAL EQUATIONS
Differential equations are used to calculate the +๐ถ term of the integral.
When solving differential equations, the derivative and a ๐ฅ and ๐ฆ value is given. To
solve the equation the derivative must be integrated, and the
๐๐ฆ
๐๐ฅ
must also be
integrated. This will give the general solution which means this is the answer to the
differential equation at any point in the graph. To solve for the +๐ถ value the ๐ฆ
coordinate given must be subbed into the y equals in the general solution and the ๐ฅ
value must be subbed into the integral. From there the equation should be
rearranged to reveal what ๐ถ equals. Now the +๐ถ value can be put back into the
general solution to give what is called the particular solution.
Functions have restrictions on their domain (possible numbers that can be input into
a function). There are two reasons a function may have a restriction:
- Zero on the denominator.
- Positive root of a negative number.
To find the restriction on a domain, identify the part that may be equal to zero or be a
negative number and set it equal to zero, to find what ๐ฅ cannot be. Then the
restriction must be expressed as ๐ฅ โฎ โ: ๐ฅ โ .
GRAPH TRANSFORMATIONS
At Higher, graphs of functions can be altered or distorted by various โgraph
transformations.โ It is important to be able to sketch these transformations.
The graph transformations are as follows:
- ๐(๐ฅ) ยฑ ๐ฅ: this means the ๐ฆ coordinate adds or subtracts, moving the graph up
or down.
- ๐(๐ฅ ยฑ ๐): this means that if it's + the ๐ฅ coordinate subtracts, and the graph
moves left. If itโs โ the ๐ฅ coordinate adds, and the graph moves right.
- โ๐(๐ฅ): means the ๐ฆ coordinate changes sign and results in the ๐ฅ axis being
flipped.
: means the ๐ฅ coordinate changes sign and results in the ๐ฆ axis being
flipped.
: multiplies the ๐ฆ coordinate by, which results in a stretched or
compressed ๐ฆ axis
- ๐(๐๐ฅ): the ๐ฅ coordinate divides by ๐ , which compresses or stretches the ๐ฅ axis.
TRIG GRAPH TRANSFORMATIONS
As taught as National 5, there are two key trigonometric graphs; the cos graph and
sine graph. They have the general form of ๐ฆ = ๐ sin ๐ ๐ฅ + ๐ or ๐ฆ = ๐ cos ๐ ๐ฅ + ๐.
A trig graph can have four transformations:
- ๐ฆ = sin(๐ฅ โ ๐) ๐๐ ๐ฆ = cos(๐ฅ โ ๐): this affects how it shifts left or right and moves
the opposite way to the sign in the brackets.
- ๐ฆ = sin ๐ฅ + ๐ ๐๐ ๐ฆ = cos ๐ฅ + ๐: this affects if it moves up or down.
- ๐ฆ = ๐ sin ๐ฅ ๐๐ ๐ฆ = ๐ cos ๐ฅ: this affects the amplitude of the graph (as in how
high it goes)
- ๐ฆ = sin ๐ ๐ฅ ๐๐ ๐ฆ = cos ๐ ๐ฅ: this affects the period of the graph (how many sin or
cos waves there are within 360ยฐ.
LOGARITHMIC AND EXPONENTIAL GRAPH
TRANSFORMATION
Functions which are in the form ๐
๐ฅ
are known as exponential functions and
have graphs given in the form of ๐ฆ = ๐
๐ฅ
๐คโ๐๐๐ ๐ฅ > 1 or ๐ฆ = ๐
๐ฅ
๐คโ๐๐๐ 0 < ๐ < 1. an
exponential graph always goes through the points (0,1) and (1,a).
In exponential graph questions, it can be asked to find the unknown coordinate by
subbing in known coordinates.
Logarithms (logs) are the inverse of exponentials and so log graphs are the inverse of
exponential graphs. This also means that log graphs pass through the points (1,0)
and (a,1).
Two key logs that must be memorised are:
๐
1 = 0 which is the log of the exponential ๐
0
๐
๐ = 1 which is the log of the exponential ๐
1
In log graph questions, it can be asked to find the unknown coordinate by subbing in
known coordinates as well.
