Year 1 - Pure Flashcards
4.5 What does the transformation of y = f(x + a) do?
Translates a by the column vector (-a,0)
4.5 What does the transformation of y = f(x) + a do?
Translates a by the column vector (0, a)
4.5 What happens to any asymptotes when a function is translated?
The asymptote is also translated
4.6 What does multiplying a constant outside a function do?
e.g. y = af(x)
Stretches y = f(x) by a scale factor a in the vertical direction (multiply the y coordinate by a)
4.6 What does multiplying a constant inside a function do?
e.g. y = f(ax)
Stretches the graph y = f(x) by a scale factor of 1/a in the horizontal direction (all x coordinates are multiplied by 1/a)
4.6 What does y = f(-x) do to the graph of y = f(x)?
Reflects the graph in the y-axis
4.6 What does y = -f(x) do to the graph of y = f(x)
Reflects the graph in the x-axis
5.1-5.4 How do you find the distance between two given points?
sqrt((Δx)^2 + (Δy)^2)
5.1-5.4 What do the equations of 2 parallel lines have in common?
They have the same gradient
5.1-5.4 How are the equations of 2 perpendicular lines different from each other?
Their gradients are negative reciprocals of each other (their product is -1)
5.1-5.4 What is the point slope formula?
y-y1=m(x-x1)
6.1 What is the midpoint of a line segment?
M((x1+x2)/2 , (y1+y2)/2)
6.2 What is the general formula for a circle which has a centre not at the origin?
(x-a)^2 + (y-b)^2 = r^2
Centre (a,b), radius r
6.3 Describe the different situations where a line intersects a circle (reference the discriminant)
2 intersections - b^2 -4ac > 0
Tangent - b^2 -4ac = 0
No intersections - b^2 -4ac < 0
6.4 What is the property of a tangent of a circle relating to the radius?
A tangent is always perpendicular to the radius at the point of intersection
6.4 What is the property of a chord relating to a circle?
The perpendicular bisector of a chord of a circle will always go through the centre of the circle
6.5 What is a circumcircle?
A circle where the vertices of a triangle are on the circumference
6.5 What is the centre of a circumcircle called and what property does it have relating to the sides of the triangle?
Circumcentre - it is the point where the perpendicular bisectors of the sides of the triangle intersect
6.5 What is the circle theorem relating to a semi circle?
The angle subtended by the diameter of the circle makes a right angle at the circumference
7.3 What is the factor theorem?
If f(p)=0 is a root of a polynomial, then (x-p) is a factor of the polynomial
OR
If (x-p) is a factor of the polynomial, then f(p)=0
7.4 What are the methods of proof?
By deduction, by exhaustion, by counter example, by contradiction
8.1 Which row of Pascal’s triangle do you use to find the coefficients for the expansion of (a+b)^n ?
(n+1)th row
8.2 What does n! mean?
n x (n-1) x (n-2) x … x 1
8.2 How do you write the number of ways of choosing r items from a group of n items?
nCr ‘n choose r’
OR
(n over r) in vector form