HSC Mathematics Standard 2 flashcards

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

Gradient formula

For two points (x1, y1) and (x2, y2), the gradient is m = (y2 − y1)/(x2 − x1). It measures vertical change per unit of horizontal change.

y-intercept

The value of y where a graph crosses the vertical axis, found by substituting x = 0 into the equation of the line.

Axis of symmetry

For the parabola y = ax² + bx + c, the vertical line x = −b/(2a) that divides the parabola into two mirror-image halves.

Discriminant

The expression b² − 4ac from the quadratic formula. If positive there are two real roots, if zero one repeated root, if negative no real roots.

Simultaneous equations

Two or more equations solved together; the solution is the set of values satisfying all equations at once, found by substitution or elimination.

Exponential growth formula

A quantity modelled by A = P(1 + r)ⁿ, where P is the starting amount, r the rate per period as a decimal, and n the number of periods.

Circumference

The perimeter of a circle, given by C = 2πr or C = πd, where r is the radius and d is the diameter.

Sector area

The area of a pie-shaped part of a circle, calculated as (θ/360) × πr² when the sector angle θ is measured in degrees.

Trapezium area

Area = ½(a + b)h, where a and b are the lengths of the parallel sides and h is the perpendicular distance between them.

Prism volume

The volume of any prism equals the cross-sectional area multiplied by its length, because the same shape is repeated along the length.

Cone volume

Volume of a cone is one third of the volume of the matching cylinder: V = ⅓πr²h, where h is the perpendicular height.

Sphere surface area

The total outside area of a sphere is S = 4πr², equal to four times the area of a circle with the same radius.

Similar solids scaling

If corresponding lengths of two similar solids are in ratio k, their surface areas are in ratio k² and their volumes in ratio k³.

Simple interest

Interest calculated only on the original principal using I = PRN, where R is the annual rate as a decimal and N the time in years.

Compound interest formula

A = P(1 + r)ⁿ gives the total amount after n periods, where r is the interest rate per period expressed as a decimal.

Straight-line depreciation

A fixed amount is subtracted each period, so the value is P − Dn, where D is the annual depreciation and n the number of years.

Declining-balance depreciation

Value falls by a fixed percentage each period, modelled by A = P(1 − r)ⁿ, where r is the decimal rate of depreciation.

GST calculation

Australian GST is 10%, so the GST-inclusive price is the original price × 1.1, and the GST component equals the inclusive price divided by 11.

Compounding frequency change

For interest compounded more than once a year, divide the annual rate by the periods per year and multiply the years by the same number.

Median

The middle score of a data set arranged in order. With an even number of scores it is the average of the two middle scores.

Interquartile range

IQR = Q3 − Q1, the range covering the middle 50% of ordered data. It is less affected by extreme values than the range.

Outlier rule

A score is treated as an outlier if it lies below Q1 − 1.5 × IQR or above Q3 + 1.5 × IQR.

Standard deviation

A measure of how far scores typically spread from the mean. Larger values indicate greater variability within the data set.

z-score

z = (x − μ)/σ gives the number of standard deviations a score lies above or below the mean; negative z means below.

Box plot features

A box plot marks the minimum, first quartile, median, third quartile and maximum, showing spread and skewness of a data set.

Correlation coefficient

A number r between −1 and 1 describing the strength and direction of linear association; values near zero mean little linear relationship.

Vertex degree

The degree of a vertex is the number of edges meeting it. Summing all degrees in a network gives twice the number of edges.

Eulerian circuit condition

A closed trail using every edge exactly once exists only when the network is connected and every vertex has an even degree.

Spanning tree

A connected sub-network containing every vertex of the original network and no cycles, so it has one fewer edge than vertices.

Kruskal's algorithm

To find a minimum spanning tree, repeatedly choose the cheapest available edge that does not create a cycle, until all vertices are connected.

Shortest path

The route between two vertices with the smallest total edge weight, found using Dijkstra's algorithm or by comparing possible paths.

Adjacency matrix

A grid where the entry in row i and column j records the number of edges joining vertex i to vertex j in the network.