Pure 1 Flashcards
a^m x a^n =
a^m+n
a^m / a^n =
a^m-n
(a^m)^n =
a^mn
(ab)^n =
a^n b^n
root(ab) =
root(a) x root(b)
root(a/b) =
root(a) / root(b)
quadratic formula
x = (-b ± root(b^2 - 4ac)) / 2a
What is the domain
The set of possible inputs for a function
What is the range
The set of possible outputs for a function
What are the roots of a function?
The values of x for which f(x) = 0
if the completed square was: f(x) = (x+p)^2 + q what would the turning point be?
(-p, q)
What is the discriminant?
b^2 - 4ac
How many roots does b^2 - 4ac > 0 have?
Two distinct real roots
How many roots does b^2 - 4ac = 0 have?
One repeated root
How many roots does b^2 - 4ac < 0 have?
No real roots
What is x is greater than y or x is smaller than or equal to b in set notation?
{x: x is greater than y} u {x: x is smaller than or equal to b}
What is x is greater than y and x is smaller than or equal to b?
{x: y is smaller than x which is smaller than or equal to b}
Is the circled filled in or empty for x is greater than or equal to y?
Filled in
Is the circled filled in or empty for x is greater than y?
Empty
Steps to solve a quadratic inequality
.Solve like a normal quadratic
.Draw a graph
.If solving for ax^2 + bx + c is greater than 0, then the answer is the x values above the x axis
.If solving for ax^2 + bx + c is smaller than 0, then the answer is the x values below the x axis
Is y is greater than f(x) dotted or solid?
Dotted
Is y greater than or equal to f(x) dotted or solid?
Solid
and
n
or
u
if x^3 is positive where does the graph start?
Bottom left
if x^3 is negative where does the graph start?
Top left
f(x) + a
Vertical by a
f(x + a)
Horizontal by -a
a X f(x)
Stretch by factor of a vertically
f(a X x)
Stretch by a scale factor of 1/a horizontally
-f(x)
Reflection of f(x) in x axis
f(-x)
Reflection of f(x) in the y axis
formula for m
(y2 - y1) / (x2 - x1)
formula for finding equation of a straight line
y - y1 = m(x - x1)
What do parallel lines have that are the same?
Gradient
How are the gradients of two perpendicular lines found?
Their gradients will multiply to make -1
How to find length of straight line between two points?
Pythagoras’ formula
How to find mid point of two points
((x1 + x2) / 2, (y1 + y2) / 2)
Where does a perpendicular bisector pass through?
The mid point
The equation of a circle with centre (0,0) and radius = r is?
x^2 + y^2 = r^2
The equation of a circle with centre (a,b) and radius = r is?
(x - a)^2 + (y - b)^2 = r^2
What is a tangents relationship with the radius?
Perpendicular
What will the perpendicular bisector of a chord do?
Go through the centre of a circle
The angle in a semicircle is always a…
Right angle
What does it mean to prove a mathematical statement true by exhaustion?
Breaking the statement into smaller cases and proving each case separately
What does it mean to prove a mathematical statement not true by counter-example?
A counter-example is one example that does not work for the statement, you do not need to give more than one example, as one is sufficient to disprove a statement
What does it mean to prove a mathematical statement true by deduction?
Starting from known factors or definitions, then using logical steps to reach the desired conclusion
What is the general formula for binomial expansion? Using (a+b)^n as an example
a^n + nC1a^(n-1)b + nC2a^(n-2)b^2 + nC3a^(n-3)b^3 + … + b^n
Cosine rule for a missing side
a^2 = b^2 + c^2 - 2bc cos(A)
Cosine rule for a missing angle
cos(A) = (b^2 + c^2 - a^2) / 2bc
Sine rule for finding missing side
a / sin(A) = b / sin(B) = c / sin(C)
Sine rule for finding missing angle
sin(A) / a = sin(B) / b = sin(C) / c
Trigonometric formula for area of a triangle
area = 1/2 ab sin(C)
sin^2(x) + cos^2(x) =
1
sin(x) / cos(x)
tan(x)
A vector has both magnitude and direction? T or F
T
If PQ–> = RS–> then
The line segments PQ and RS are equal in length and are parallel
If AB–> = -BA–> then
AB is equal in length, parallel and in the opposite direction to BA
AB–> + BC–> =
AC–>
What is lamda?
A non-zero scalar
What is a unit vector? How are they usually denoted?
A vector of length 1, usually denoted by i and j, for x and y respectively
How would you find the magnitude of: a = xi + yj
[a] = sqrt(x^2 + y^2)
What are position vectors?
Vectors giving the position of a point, relative to a fixed origin
Equation for differentiating from first principles
f’(x) = lim of h→0 of (f(x+h) - f(x)) / (h)
How do you find the derivative usually?
Times by the power and then minus one from the power
if f’‘(a) > 0 then
The point is a local minimum
if f’‘(a) < 0 then
The point is a local maximum
Generally how do you integrate?
Add one to the power and then divide the coefficient by the value of the new power
If not integrating to find an area on a graph, what must you remember to do?
Constant of integration = +c
How must normal integration be set out?
Integration squiggle, equation wanting to integrate, dx
Where do you put the limits on definite integrals?
At the top and bottom of the squiggly line
Which limits go where?
Highest value on the top
Which limit value gets taken away from which one?
Top minus bottom
Stages of definite integrals?
Squiggly line, square bracket, two normal brackets
When integrating to find area, what area does it find? If it’s under the x-axis what must you do?
The area between the curve and the x-axis, if it is under the x-axis it will be negative so you need to flip it
Why is e special?
The derivative of e^x = e^x
What is a^x = n as a log?
log(a)n = x
log x + log y =
log xy
log x - log y =
log x/y
log x^k =
k * log x
ln x =
log(e) x
e^(ln x) =
ln e^x = x
How can you use logs to understand non-linear data?
Mess around with the equation till you have a linear equation with logs, e.g. (log y = n * log x + log a) is the same as y = mx + c
If y = ax^n then the graph of log y against log x will be
A straight line with gradient n and vertical intercept log a
If y = an^x then the graph of log y against x will be
A straight line with gradient log n and vertical intercept log a
