algebra 1 Flashcards
(20 cards)
proof
logical argument for a mathematical statement. Shows that something must be either true or false
direct proof
- assume that a statement, P, is true
- use P to show that another statment, Q, is true
proof by exhaustion
- list a set of cases that exhausts all possibilities
- show that the statement is true in each and every case
index laws
- any number raised to the power zero is 1
- negative powers may be written as reciprocals eg x^-n= 1/x^n
-any base raised to the power of a unit fraction is a root eg x^1/n= 1/rootx
-when multiplying terms you add the indices - to divide terms you subtract the indicies
- to raise one term to another power you multiply the indicies
what is a rational number
one that you can write exactly in the form p/q where p and q are integers and q isn’t 0
surd laws
√a x √b = √ab
√a /√b= √ a/b
you can also use the difference of two squares to simplify a fraction with a surd at the bottom to make it more rational and put the surd on the top
completing the square
ax^2 + bx +c = a(x+b/2a) ^2 +q
quadratic formula
x= −b±√b2−4ac
————–
2a
discriminant
b^2- 4ac
if discriminant >0 the quadratic has 2 distinct roots and the curve crosses the x-axis at two distinct points
if discriminant =0 the quadratic has one repeated root and the x-axis is a tangent to the curve at this point
if discriminant <0 the quadratic has no real roots and the curve doesnt cross the x-axis at any point
ways of solving simultaneous equations
- graphically
-by eliminating one of the variables - substitution
ways of writing equation of a straight line
y-y1= m(x-x1)
where (x1, y1)is a point on the line
gradient of a straight line
y2-y1
……………….
x2-x1
distance between two points
d=√((x_2-x_1)²+(y_2-y_1)²
midpoint of the line
x_1 +x_2 , y_1 +y_2
———— ————-
2 2
equation of circle
(x-a) ^2 + (y-b)^ 2 = r^2
circle theorems
- if a triangle passes through the centre of the circle, and all three corners touch the circumference of the circle, then the triangle is right- angles
- the perpendicular line from the centre of the circle to a chord bisects the chord
- any tangent to a circle is perpendicular to the radius at the point of contact (B)
ways of representing inequalities
you can represent inequalities on a number line.
You use a coloured/ shaded circle when representing greater or equal to, or less than or equal to
you use an empty circle when representing > or <
- interval notation
- set notation
when you divide or multiply by negative number inequality sign flips
set notation
uses curly brackets and the intersection and union symbols
eg -1< x ≤ 4
can be written as:
{x: x> -1 } ∩ {x : x ≤ 4 }
can be simplified to
{ x: -1 <x ≤4}
interval notation
uses square and round brackets
[,] for weak inequalities of ≤ or ≥
(,) for < or >
infinity or negative infinity can be used with round brackets to show the upper or lower bound
- union symbol can be used to show two separate inequalities
Interval notation uses different brackets to indicate whether a number is included or not
- [ or ] mean included
- ( or ) mean excluded
- (4,8] means 4 < x < 8
- Note ∞ always uses ( or )
partial fraction rules
- if denominator contains a squared factor you need to consider the squared factor as a possible denominator, as well as all linear factors- so would lead to two separate partial fractions plus any others
- if the order of the numerator is equal to or greater than the order of the denominator, you divide the numerator by the denominator first and then express the remainder in partial fractions. So would need to have an individual constant term that would not be in partial fraction form but needs to be calculated
