SECTION 2- ALGEBRA AND FUNCTIONS

Question 1

INDICES RULE - multiplication

Answer 1

if you multiply the number, you add the powers

- bases MUST be the same e.g a^m x a^n = a^m+n

Question 2

INDICES RULE - division

Answer 2

if divide two numbers, you subtract their powers

- bases must be the same e.g. a^m / a^n = a^m-n

Question 3

INDICES RULES - brackets

Answer 3

if you have a power to the power of something else – multiply the powers together e.g. (a^m)^n = a^mn

Question 4

INDICES RULE - negative power

Answer 4

• a negative power means it’s on the bottom line of a fraction e.g. a^-m = 1 / a^m

Question 5

INDICES RULE - power of 0

Answer 5

• any number to the power of 0 is equal to 1 e.g. (a+b)^0 =1

Question 6

INDICES RULE - a^1 / m

Answer 6

• roots can be written as powers so: a^1/ m = m√a

Question 7

INDICES RULE - a^m/ n

Answer 7

• a power that’s a fraction is the root of a power - or the power of a root so: a^m/ n = n√a^m = (n√a)^m

Question 8

what does it mean to ‘rationalise the denominator’

Answer 8

• means getting rid of the surds from the bottom of the fraction

Question 9

how do you ‘simplify a surd’

Answer 9

• to simplify a surd, make the number in the √ sign smaller, or get rid of a fraction in the √ sign

Question 10

what are the 3 rules of surds

Answer 10

• √ab = √a√b
• √a/b = √a /√b
• (√a)^2 = √a√a

Question 11

what is the difference of two squares

Answer 11

• can be applied to surds also

- (a+b) (a-b) = a^2 - b^2

Question 12

how to simplify algebraic fractions

factorising

Answer 12

• look for common factors in the numerator and denominator – factorise top and bottom and see if there’s anything you can cancel
• if there’s a fraction in the numerator or denominator –multiply top and bottom by by the same factor

Question 13

how to add and subtract fractions by finding a common denominator

Answer 13

• find the common denominator
• put each fraction over the common denominator
• combine into one fraction

Question 14

what does the term ‘degree’ mean

Answer 14

• the highest power of x in the polynomial

Question 15

what does the term ‘divisor’ mean

Answer 15

• the thing that you are dividing by

Question 16

what does the term ‘quotient’

Answer 16

• the stuff that you get when you divide by the divisor (not including the remainder)

Question 17

what does the term ‘remainder’ mean

Answer 17

• the bit that’s left over at the end (known in A-Levels as the constant)

Question 18

what is the general quadratic equation

Answer 18

ax^2 + bx + c

Question 19

what is the quadratic formula

Answer 19

x= −b±√b²−4ac /2a

Question 20

what is the discriminant

Answer 20

b²-4ac

Question 21

what happens when the discriminant is

a) =0
b) <0
c) >0

Answer 21

a) when it’s equal to zero, it has one repeated root
b) when it’s less than zero, it has no roots (neither of the solutions are real numbers)
c) when it’s greater than zero, it has two distinct roots

Question 22

what is the factor theorem

Answer 22

• if you get a remainder of zero when you divide the polynomial f(x) by (x - a), then (x - a) must be a factor of f(x)

Question 23

how to use the factor theorem when there’s a coefficient of x

Answer 23

• to test if (ax - b) is a factor. substitute f(b/a) and if it =0, it’s a factor
• so if you know the roots, you know the factors – and vice versa

Question 24

what does transformation y = f(x + c) mean

Answer 24

f(x) is shifted c to the left

- graph is moved in opposite direction e.g. f(x + c) shifts to the left and f(x - c) is shifted to the right

Question 25

what does transformation y = f(x) + c mean

Answer 25

• movement on y - axis, so will shift c upwards and if it’s f(x)- c, it will shift downwards in y-axis

Question 26

what does transformation y = af(x) mean

Answer 26

• is stretched in y direction w/ scale factor of a
• if |a| > 1 the graph of af(x) is f(x) stretched vertically by a factor of a
• if 0< |a| < 1 the graphed is squashed vertically
• if a a < 0 the graph is also reflected in the x-axis

Question 27

what does transformation y = f(ax) mean

Answer 27

• is stretched in x direction w/ scale factor of 1/a
• if |a| > 1, the graph of f(ax) is f(x) squashed horizontally by a factor of a
• if 0 < |a| < 1, the graph is stretched horizontally
• if a < 0, the graph is also reflected in the y-axis

Question 28

what does transformation y = -f(x)

Answer 28

• a reflection in x-axis

Question 29

what does transformation y = f(-x)

Answer 29

• a reflection in y-axis