1 - Algebra and Functions Flashcards
What is a set?
a collection of objects or numbers
How is a set shown?
{ }
What does p ⇒ q mean?
p implies q
What does p ⇔ q mean?
p implies q and q implies p
What do you have to do in proof by exhaustion?
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
What do you have to do in proof by deduction (direct proof)?
use known facts to build up argument
What do you have to do in disproof by counter-example?
find one case where the statement doesn’t hold
aᵐ x aⁿ =
aᵐ⁺ⁿ
aᵐ / aⁿ =
aᵐ⁻ⁿ
a¹/ᵐ
ᵐ√a
aᵐ/ⁿ
ⁿ√aᵐ
a⁻ᵐ
1/aᵐ
a⁰
1
When you divide f(x) by (x-a), what is the remainder?
f(a)
When you divide f(x) by (ax-b), what is the remainder?
f(b/a)
If the coefficient of x² is positive, what is the shape of the graph?
u-shaped
If the coefficient of x² is negative, what is the shape of the graph?
n-shaped
What is the discriminant?
b² - 4ac
What does the discriminant tell you?
if a quadratic equation has:
1) two roots
2) one real root
3) no real roots
What does b² - 4ac > 0 tell you?
two real roots
What does b² - 4ac = 0 tell you?
one real root
What does b² - 4ac < 0 tell you?
no real roots
What happens when multiplying or dividing an inequality by negative numbers?
the direction of the inequality sign changes
What does {x: x<a></a>
the set of values of x such that x is less than a
What is the union (∪) of two sets?
everything contained in either set
What is the intersection (∩) of two sets?
the things only present in both sets
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
If the coefficient of x³ is positive, what is the shape of the graph?
bottom-left to top-right
If the coefficient of x³ is negative, what is the shape of the graph?
top-left to bottom-right
For f(x+c), what happens to f(x)?
shifted c to the left
For f(x-c), what happens to f(x)?
shifted c to the right
For f(x) + c, what happens to f(x)?
shifted c upwards
For f(x) - c, what happens to f(x)?
shifted c downwards
Which way do reflections in the x-axis flip f(x)?
vertically
Which way do reflections in the y-axis flip f(x)?
horizontally
What is the equivalent of y ∝ x?
y = kx
What is the equivalent of y ∝ 1/x?
y = k/x
