HSC Maths Advanced flashcards

31 cards. Read the term and the definition here. The deck at the top of this page is optional.

Vertical line test

A graph represents a function only if no vertical line meets it more than once, meaning each input value has exactly one output value.

Composite function domain

For f(g(x)), the input must be in the domain of g, and the resulting g(x) must lie in the domain of f.

Inverse function condition

An inverse exists only when the function is one-to-one, since each output must trace back to a single input; its graph reflects in y=x.

Odd and even functions

A function is odd if f(−x)=−f(x), giving point symmetry about the origin, and even if f(−x)=f(x), giving symmetry about the y-axis.

Discriminant of a quadratic

For ax²+bx+c, the discriminant b²−4ac determines the number of real roots: two if positive, one if zero, none if negative.

Quadratic axis of symmetry

The parabola y=ax²+bx+c has axis of symmetry x=−b/(2a), and the vertex lies on this line at the turning point.

Absolute value equation method

To solve |expression|=k with k positive, set expression=k and expression=−k separately, solve each, then verify both results satisfy the original.

Sine, cosine, tangent ratios

In a right triangle, sine is opposite over hypotenuse, cosine is adjacent over hypotenuse, and tangent is opposite over adjacent.

Pythagorean trigonometric identity

sin²θ+cos²θ=1 holds for every angle θ, and dividing throughout by cos²θ yields 1+tan²θ=sec²θ.

ASTC quadrant rule

In quadrants one to four, All, Sine, Tangent, Cosine respectively are positive; the related acute angle generates the second solution.

Amplitude and period

For y=a sin(bx)+d, amplitude is |a| and period is 360°/|b| in degrees, or 2π/|b| in radians.

Sine rule

a/sin A = b/sin B = c/sin C applies in any triangle and is used when a side and its opposite angle are known.

Cosine rule

c²=a²+b²−2ab cos C finds a side or angle in any triangle, especially when no matching side-angle pair is given.

Triangle area with sine

The area of any triangle equals half the product of two sides multiplied by the sine of the included angle: ½ab sin C.

Arc length in radians

When the central angle θ is measured in radians, arc length is l=rθ and sector area is A=½r²θ.

Derivative first principles

The derivative is the limit as h approaches zero of [f(x+h)−f(x)]/h, representing the instantaneous rate of change.

Chain rule

The derivative of f(g(x)) equals f′(g(x)) multiplied by g′(x), used for composite functions such as (2x+1)⁵.

Product rule

For a product uv, the derivative is u′v+uv′, differentiating one factor at a time while holding the other constant.

Quotient rule

For u divided by v, the derivative is (u′v−uv′)/v², valid wherever v is not zero.

Stationary point classification

A stationary point has f′(x)=0; the second derivative being positive means a minimum, negative a maximum, and zero requires a sign test.

Definite integral evaluation

The definite integral from a to b equals F(b)−F(a), where F is any antiderivative of the integrand; no constant of integration is needed.

Trapezoidal rule

Numerical integration approximating area by trapezoids: (h/2) times the sum of the first and last ordinates plus twice all interior ordinates.

Compound interest formula

A=P(1+r)ⁿ gives the amount after n periods at decimal rate r per period, with interest compounding each period.

Reducing balance depreciation

Value after n periods is S=V(1−r)ⁿ, where r is the decimal rate of decrease applied to the remaining value each period.

Future value of annuity

FV=M[(1+r)ⁿ−1]/r gives the accumulated value of n regular payments M made at rate r per period.

Present value of annuity

PV=M[1−(1+r)⁻ⁿ]/r is the single amount today equivalent to n future payments of M at periodic rate r.

Effective annual rate

When nominal rate i compounds m times yearly, the effective annual rate is (1+i/m)^m−1, allowing fair comparison of investments.

Interquartile range and outliers

IQR=Q₃−Q₁ measures the spread of the middle half; values beyond 1.5 times the IQR from the quartiles are outliers.

Standard deviation meaning

Standard deviation is the typical distance of data values from the mean; a larger value indicates greater spread.

z-score

A z-score of (x−x̄)/s states how many standard deviations a value lies from the mean, enabling comparison across different datasets.

Correlation coefficient

The value r between −1 and 1 measures the strength and direction of linear association between two variables.