QCAA Maths methods External focus

covers all topics in the qcaa external

Function

A rule that assigns exactly one output for each input

Domain of a function

The set of all possible input values for which the function is defined

Range of a function

The set of all possible output values the function can produce

Rule of a function

The algebraic expression that defines how inputs are mapped to outputs

Difference between a relation and a function

A relation can assign multiple outputs to one input while a function assigns exactly one output to each input

Test if a relation is a function

Use the vertical line test; if any vertical line cuts the graph more than once it is not a function

One to one function

Each input value has a unique output and each output has only one input

Inverse function

A function that reverses the mapping of the original function

How to find the inverse function

Swap x and y in the equation and solve for y

Domain of an inverse function

It is the range of the original function

Range of an inverse function

It is the domain of the original function

Composition of functions

Applying one function to the result of another written as f of g of x

Function composition notation

f circle g of x equals f of g of x

Condition for valid composition

The range of the inner function must fit within the domain of the outer function

Piecewise function

A function defined by different rules over different intervals of the domain

Transformation of a function

A change to the graph by shifting stretching reflecting or compressing it

Translation of a graph

Moves the graph without changing its shape or orientation

Effect of f(x) - a on graph

Shifts the graph a units to the right

Effect of f(x) + a on graph

Shifts the graph a units to the left

Effect of f(x) + k on graph

Shifts the graph vertically upward by k units

Effect of f(x) - k on graph

Shifts the graph vertically downward by k units

Reflection in the x axis

Multiply function by -1 to reflect over x axis

Reflection in the y axis

Replace x by -x to reflect over y axis

Horizontal stretch or compression

Changes the width of the graph by multiplying x by a constant factor

Vertical stretch or compression

Changes the height of the graph by multiplying the function value by a constant factor

Effect of multiplying f(x) by a constant > 1

Graph becomes vertically stretched

Effect of multiplying f(x) by a constant between 0 and 1

Graph becomes vertically compressed

Equality of two functions

They produce the same output for all x in their common domain

Asymptotes

Lines that a graph approaches but never touches

Types of asymptotes

Vertical horizontal and oblique

Vertical asymptote in rational function

Occurs when the denominator approaches zero

Horizontal asymptote

Occurs when the function approaches a constant as x increases or decreases

Oblique asymptote

Occurs when the degree of numerator is one greater than denominator

Shape of a quadratic function

A parabola that opens upward or downward

Vertex of a parabola

Point representing the maximum or minimum value of the function

Finding vertex of y = ax^2 + bx + c

Use x = -b/2a and substitute to find y

Axis of symmetry

Vertical line passing through the vertex dividing the graph into two equal halves

Determine if quadratic opens up or down

If a is positive opens up, if negative opens down

Exponential function

A function where the variable is in the exponent, typically written as f(x) = a^x

Base of an exponential function

The constant a in a^x, must be positive and not equal to 1

Domain of an exponential function

All real numbers

Range of an exponential function

For a positive base, all positive real numbers

Shape of exponential function graph

Increasing if base > 1, decreasing if 0 < base < 1

Solving basic exponential equations

Rewrite both sides with the same base and equate exponents

Natural exponential function

The function e^x, where e ≈ 2.718

Derivative of e^x

e^x

Derivative of a^x

a^x * ln(a)

Integral of e^x dx

e^x + C

Integral of a^x dx

a^x / ln(a) + C

Logarithmic function

The inverse of an exponential function

Natural logarithm

Logarithm with base e, written as ln

Domain of a logarithmic function

All positive real numbers

Range of a logarithmic function

All real numbers

Derivative of ln x

1 / x

Integral of 1/x dx

ln|x| + C

Solving logarithmic equations

Rewrite in exponential form or combine using log properties

Product rule for logarithms

log(AB) = log A + log B

Quotient rule for logarithms

log(A/B) = log A - log B

Power rule for logarithms

log(A^n) = n * log A

Exam tip for logs and exponents

Always check solutions in the original equation to avoid extraneous answers

Graph transformations for exponentials/logs

Shifts, reflections, stretches, and compressions

Horizontal asymptote for basic exponential function

y = 0

Solving equations with different bases

Use logarithms on both sides to bring exponents down

Identifying growth or decay

Base > 1 is growth, 0 < base < 1 is decay

Derivative of ln(u)

1 / u * du/dx

Derivative of e^u

e^u * du/dx

Exam tip for sketching exponential graphs

Identify base, asymptote, and transformations to quickly sketch

Trigonometric function

A function that relates angles of a triangle to ratios of sides or to the unit circle

Primary trigonometric functions

Sine, cosine, and tangent

Reciprocal trigonometric functions

Cosecant, secant, and cotangent

Domain of sine and cosine functions

All real numbers

Range of sine and cosine functions

From -1 to 1 inclusive

Domain of tangent function

All real numbers except where cosine = 0

Range of tangent function

All real numbers

Period of sine and cosine functions

2 pi

Period of tangent function

Pi

Amplitude of sine or cosine function

The maximum value minus the minimum value divided by 2, or absolute value of the coefficient

General solution for sine equations

Use sin θ = k, solve for θ within one period, then add n*2π

General solution for cosine equations

Use cos θ = k, solve for θ within one period, then add n*2π

General solution for tangent equations

Use tan θ = k, solve within one period, then add n*π

Derivative of sin x

Cos x

Derivative of cos x

• Sin x

Derivative of tan x

Sec^2 x

Integral of sin x dx

• Cos x + C

Integral of cos x dx

Sin x + C

Integral of sec^2 x dx

Tan x + C

Pythagorean identity sin and cos

Sin^2 x + Cos^2 x = 1

Pythagorean identity tan and sec

Tan^2 x + 1 = Sec^2 x

Pythagorean identity cot and csc

1 + Cot^2 x = Csc^2 x

Transformations of trigonometric graphs

Shifts, reflections, stretches, and compressions

Amplitude exam tip

Absolute value of coefficient before sine or cosine

Period calculation

2 pi divided by coefficient of x for sine/cosine, pi for tangent

Exam tip for solving trig equations

Check domain and consider all solutions in the interval

Derivative of sin(u)

Cos(u) * du/dx

Derivative of cos(u)

• Sin(u) * du/dx

Derivative of tan(u)

Sec^2(u) * du/dx

Integral exam tip

Look for substitution or standard formulas; check limits for definite integrals

Degrees to radians conversion

Degrees * pi / 180