1 - Algebra and Functions

Question 1

What is a set?

Answer 1

a collection of objects or numbers

Question 2

How is a set shown?

Answer 2

{ }

Question 3

What does p ⇒ q mean?

Answer 3

p implies q

Question 4

What does p ⇔ q mean?

Answer 4

p implies q and q implies p

Question 5

What do you have to do in proof by exhaustion?

Answer 5

1) break things down into two or more cases
2) make sure cases cover all possible situations
3) separately prove that the statement is true for each case

Question 6

What do you have to do in proof by deduction (direct proof)?

Answer 6

use known facts to build up argument

Question 7

What do you have to do in disproof by counter-example?

Answer 7

find one case where the statement doesn’t hold

Question 8

aᵐ x aⁿ =

Answer 8

aᵐ⁺ⁿ

Question 9

aᵐ / aⁿ =

Answer 9

aᵐ⁻ⁿ

Question 10

a¹/ᵐ

Answer 10

ᵐ√a

Question 11

aᵐ/ⁿ

Answer 11

ⁿ√aᵐ

Question 12

a⁻ᵐ

Answer 12

1/aᵐ

Question 13

a⁰

Answer 13

1

Question 14

When you divide f(x) by (x-a), what is the remainder?

Answer 14

f(a)

Question 15

When you divide f(x) by (ax-b), what is the remainder?

Answer 15

f(b/a)

Question 16

If the coefficient of x² is positive, what is the shape of the graph?

Answer 16

u-shaped

Question 17

If the coefficient of x² is negative, what is the shape of the graph?

Answer 17

n-shaped

Question 18

What is the discriminant?

Answer 18

b² - 4ac

Question 19

What does the discriminant tell you?

Answer 19

if a quadratic equation has:

1) two roots
2) one real root
3) no real roots

Question 20

What does b² - 4ac > 0 tell you?

Answer 20

two real roots

Question 21

What does b² - 4ac = 0 tell you?

Answer 21

one real root

Question 22

What does b² - 4ac < 0 tell you?

Answer 22

no real roots

Question 23

What happens when multiplying or dividing an inequality by negative numbers?

Answer 23

the direction of the inequality sign changes

Question 24

What does {x: x<a></a>

Answer 24

the set of values of x such that x is less than a

Question 25

What is the union (∪) of two sets?

Answer 25

everything contained in either set

Question 26

What is the intersection (∩) of two sets?

Answer 26

the things only present in both sets

Question 27

How do you factorise a polynomial when you haven’t been given any factors?

Answer 27

1) try small numbers until f(something) = 0
2) use the factor theorem
3) write down another set of brackets

Question 28

If the coefficient of x³ is positive, what is the shape of the graph?

Answer 28

bottom-left to top-right

Question 29

If the coefficient of x³ is negative, what is the shape of the graph?

Answer 29

top-left to bottom-right

Question 30

For f(x+c), what happens to f(x)?

Answer 30

shifted c to the left

Question 31

For f(x-c), what happens to f(x)?

Answer 31

shifted c to the right

Question 32

For f(x) + c, what happens to f(x)?

Answer 32

shifted c upwards

Question 33

For f(x) - c, what happens to f(x)?

Answer 33

shifted c downwards

Question 34

Which way do reflections in the x-axis flip f(x)?

Answer 34

vertically

Question 35

Which way do reflections in the y-axis flip f(x)?

Answer 35

horizontally

Question 36

What is the equivalent of y ∝ x?

Answer 36

y = kx

Question 37

What is the equivalent of y ∝ 1/x?

Answer 37

y = k/x