1 - Algebra and Functions Flashcards

1
Q

What is a set?

A

a collection of objects or numbers

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2
Q

How is a set shown?

A

{ }

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3
Q

What does p ⇒ q mean?

A

p implies q

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4
Q

What does p ⇔ q mean?

A

p implies q and q implies p

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5
Q

What do you have to do in proof by exhaustion?

A

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

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6
Q

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

A

use known facts to build up argument

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7
Q

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

A

find one case where the statement doesn’t hold

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8
Q

aᵐ x aⁿ =

A

aᵐ⁺ⁿ

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9
Q

aᵐ / aⁿ =

A

aᵐ⁻ⁿ

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10
Q

a¹/ᵐ

A

ᵐ√a

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11
Q

aᵐ/ⁿ

A

ⁿ√aᵐ

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12
Q

a⁻ᵐ

A

1/aᵐ

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13
Q

a⁰

A

1

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14
Q

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

A

f(a)

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15
Q

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

A

f(b/a)

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16
Q

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

17
Q

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

18
Q

What is the discriminant?

19
Q

What does the discriminant tell you?

A

if a quadratic equation has:

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

20
Q

What does b² - 4ac > 0 tell you?

A

two real roots

21
Q

What does b² - 4ac = 0 tell you?

A

one real root

22
Q

What does b² - 4ac < 0 tell you?

A

no real roots

23
Q

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

A

the direction of the inequality sign changes

24
Q

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

A

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

25
What is the union (∪) of two sets?
everything contained in either set
26
What is the intersection (∩) of two sets?
the things only present in both sets
27
How do you factorise a polynomial when you haven't been given any factors?
1) try small numbers until f(something) = 0 2) use the factor theorem 3) write down another set of brackets
28
If the coefficient of x³ is positive, what is the shape of the graph?
bottom-left to top-right
29
If the coefficient of x³ is negative, what is the shape of the graph?
top-left to bottom-right
30
For f(x+c), what happens to f(x)?
shifted c to the left
31
For f(x-c), what happens to f(x)?
shifted c to the right
32
For f(x) + c, what happens to f(x)?
shifted c upwards
33
For f(x) - c, what happens to f(x)?
shifted c downwards
34
Which way do reflections in the x-axis flip f(x)?
vertically
35
Which way do reflections in the y-axis flip f(x)?
horizontally
36
What is the equivalent of y ∝ x?
y = kx
37
What is the equivalent of y ∝ 1/x?
y = k/x