Year 1 Pure Facts/Rules Flashcards
What is the quadratic formula?
x=(-b±√(b^2-4ac))/2a
What is the discriminant?
b^2-4ac
How do you determine if a quadratic equation has no real roots?
The discriminant is negative
ie b^2-4ac<0
or b^2<4ac
How do you determine if a quadratic equation has equal (repeated) real roots?
The discriminant equals zero
ie b^2-4ac=0
or b^2=4ac
How do you determine if a quadratic equation has two real roots?
The discriminant is positive
ie b^2-4ac>0
or b^2>4ac
If you are sketching a cubic graph that has a positive x3 term, where does your sketch start - bottom left or top left?
Bottom left
How do we find the points at which a quadratic function crosses the x-axis?
Put y = 0 and solve the quadratic equation by factorising, completing the square or using the formula. The x-axis crossing points are the solutions (the ‘roots’) of the equation.
If we have completed the square to get
〖y= (x+3)〗^2+19
what are the coordinates of the minimum point of this quadratic function?
(-3,19)
Factorise x^2-y^2
(x-y)(x+y)
Complete the square: x^2-8x
(x-4)^2-16
What is the value of 〖64〗^(1/3)?
4
What is the value of 〖11〗^(-2)?
1/121
Simplify √50
5√2
How would you rationalise the denominator of 1/(a+√b)?
Multiply by (a-√b)/(a-√b)
How do you find the gradient of the straight line joining two points?
m= rise/run=(change in y)/(change in x)= (y_2-y_1)/(x_2-x_1 )
What method should you use to solve simultaneous equations where one is linear and one is quadratic?
Substitution
When solving an inequality, what must you do if you multiply or divide both sides by a negative number?
Turn the inequality sign around
What must you do when solving a quadratic inequality?
Draw a sketch
If the straight line L has gradient m, what is the gradient of the straight line perpendicular to L?
-1/m (ie the negative reciprocal)
If a tangent to a curve at the point P has gradient-2/3, what is the gradient of the normal to the curve at the point P?
3/2
The point A(3,5) lies on the curve y = f(x).
What are the new coordinates of the point A under the transformation y = f(-x)?
( -3 , 5 )
The point A(3,5) lies on the curve y = f(x).
What are the new coordinates of the point A under the transformation y = f(x+3)?
( 0 , 5 )
The point A(3,5) lies on the curve y = f(x).
What are the new coordinates of the point A under the transformation y = f(3x)?
( 1 , 5 )
The point A(3,5) lies on the curve y = f(x).
What are the new coordinates of the point A under the transformation y = 3f(x)?
( 3 , 15 )
The point A(3,5) lies on the curve y = f(x).
What are the new coordinates of the point A under the transformation y = f(x)+3?
( 3 , 8 )
The point A(3,5) lies on the curve y = f(x).
What are the new coordinates of the point A under the transformation y = -f(x)?
( 3 , -5 )
Write as a single log: logax - logay
loga(x/y)
How do we write tanx in terms of sinx and cosx?
tanx= sinx/cosx
What is the factor theorem?
If (x-a) is a factor of f(x) then f(a) = 0
What is the Sine Rule?
a/sinA=b/sinB=c/sinC
What is the Cosine Rule?
a^2=b^2+c^2-2bcCosA
What is the name of a line that touches a circle at one point only?
Tangent
What is the equation of a circle with centre (a,b) and radius r?
(x-a)^2+(y-b)^2=r^2
What is the magnitude of vector 3i+4j?
√(3^2+4^2 )=5
Given the position vectors of A and B, how would you find the vector (AB) ⃗?
(OB) ⃗-(OA) ⃗ or b - a
What is the coefficient of x^2 in the expression 〖5x〗^3+〖3x〗^2/4+x/2?
3/4
What notation do we use for ‘3 factorial’? How is it calculated and what is its value?
3! = 3 x 2 x 1 = 6
What notation do we use for ‘5 choose 2’? How is it calculated?
(■(5@2)) or 5C2 = 5!/2!3!
Write 2^3 = 8 in log form
log28 = 3
What does 〖sin〗^2 x+〖cos〗^2 x equal?
1
Write as a single log: logax + logay
logaxy
When is a function increasing?
When the gradient is positive
What is the value of loga1?
0
What is the value of logaa?
1
Write logaxn without an index
nlogax
What is the period of the Sine and Cosine functions?
360o
When is a function decreasing?
When the gradient is negative
Write log416 = 2 in index form
4^2 = 16
How do we find stationary points?
Find the point(s) where the gradient equals zero.
What does the second derivative tell us about the nature of a stationary point?
If the second derivative is positive, it is a minimum. If the second derivative is negative, it is a maximum. (If the second derivative is zero, it could be a minimum, maximum or point of inflexion).
How do we find the area under a curve between x = a and x = b?
Integrate the function and evaluate it between the limits a and b.
What is the midpoint of the points (a, b) and (c, d)?
((a+c)/2 ,(b+d)/2)
What is the inverse of y=e^x?
lnx
How do you find the area of a triangle?
Area= 1/2 absinC where C is the included angle
or Area= 1/2 b × h where h is the perpendicular height
