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A-Maths Term 1 - Coggle Diagram
A-Maths Term 1
Indices and Surds
INDICES RULES
a^m x a^n = a^m+n
a^m/a^n = a^m-n
(a^m)^n = a^mn
a^n x b^n = (ab)^n
a^n/b^n = (a/b)^n
a^0 = 1
a^-n = 1/a^n
a^1/n = ^n√a
a^m/n = ^n√a^m
Surds
A surd is an irrational number of the form of a root which is not a perfect square
Indices
An index number is a number which is raised to a power.
Exponential Equations
The equation has an unknown exponent (or index) and is called an exponential equation.
2^x = 64
Quadratic Functions
Graph of Quadratic Functions
Maximum points
Minimum Points
Parabola
Roots of Quadratic Equations
2 Distinct Roots
0
2 Distinct points of intersection
Roots are real and distinct
2 Equal Roots
= 0
1 point of intersection (line is a tangent)
Roots are real and equal
0 Roots
< 0
No points of intersection
Roots are no real
Discriminant = b^2 - 4ac
Completing the square
f(x) = ax^2 + bx + c can be written in the form f(x) = a(x - h)^2 + k
if a > 0 the minimum point is (h, k)
if a < 0, the maximum point is (h, k)
Logarithmic and Exponential Functions
Exponential
an equation containing a variable in index form
y = a^x
Logarithmic
an equation of the power to which a base must be raised to get a given answer
Common Logarithms
logarithm with the base 10
log_10(x) or lg
y = log_a(x)
Natural Logarithms
log_e(x) or ln
logarithm with base e or 2.71828
Laws
log_a(xy) = log_a(x)+log_a(y)
log_a(x)^m = mlog_a(x)
log_a(x/y) = log_a(x)-log_a(y)
Simultaneous Equations
Solving Linear Equations
Elimination Method
Substitution Method
Solving the equations algebraically
Factorization Method
Quadratic Formula Method
Completing the Square method