# VCE Mathematical Methods Units 1 and 2 – Functions and Graphs

1.1 Coordinate Geometry
1.2 Functions
1.3 Polynomials
1.4 Trigonometry
1.6 Exponential Graphs
1.7 Logarithmic Graphs
1.8 Exponential Growth and Decay

## Unit 1

In this area of study, students cover the graphical representation of simple algebraic functions (polynomial and power functions) of a single real variable and the key features of functions and their graphs such as axis intercepts, domain (including the concept of the maximal, natural or implied domain), co-domain and range, stationary points, asymptotic behaviour and symmetry. The behaviour of functions and their graphs is explored in a variety of modelling contexts and theoretical investigations.

This area of study includes:

• review of coordinate geometry
• functions and function notation, domain, co-domain and range, representation of a function by rule, graph and table
• use of the vertical line test to determine whether a relation is a function or not, including examples of relations that are not functions and their graphs such as $x = k, x = ay^2$ and circles in the form $(x–a)^2+(y–b)^2 = r^2$
• qualitative interpretation of features of graphs of functions, including those of real data not explicitly represented by a rule, with an approximate location of stationary points
• graphs of power functions $f(x) = x^n$ for $n \in N$ and $n \in \displaystyle \{–2, –1, 0,1,2 \}$, and transformations of these graphs to the form $y = a (x + b)^n + c \$ where $\ a, b, c \in R$ and $a \ne 0$
• graphs of polynomial functions to degree $4$ and other polynomials of higher degrees such as $f(x) = (x + 5)^2 (x–7)^ 3 + 10$
• graphs of inverse functions

## Unit 2

In this area of study, students cover graphical representation of functions of a single real variable and the key features of graphs of functions such as axis intercepts, domain (including maximal, natural or implied domain), co-domain and range, asymptotic behaviour, periodicity and symmetry.

This area of study includes:

• review of trigonometry (sine and cosine rules not required)
• the unit circle, radians, arc length and conversion between radian and degree measures of angle
• sine, cosine and tangent as functions of a real variable, and the relationships $\sin(x) \approx x$ for small values of $x, \sin^2(x) + \cos^2(x) = 1$ and $\tan(x) = \displaystyle \frac{\sin(x)}{\cos(x)}$
• exact values for sine, cosine and tangent of $\displaystyle \frac{n \pi}{6}$ and $\displaystyle \frac{n \pi}{4}, n \in Z$
• symmetry properties, complementary relations and periodicity properties for sine, cosine and tangent functions
• circular functions of the form $y = af (bx) + c$ and their graphs, where $f$ is the sine, cosine or tangent function, and $a, b, c \in R$ with $a, b \ne 0$
• simple applications of sine and cosine functions of the above form, with examples from various modelling contexts, the interpretation of the period, amplitude and mean value in these contexts and their relationship to the parameters $a, b$ and $c$
• exponential functions of the form $f : R \rightarrow R, f(x) = Aa^{kx} + C$ and their graphs, where $a \in R^+ , A, k, C \in R, A \ne 0$
• logarithmic functions of the form $f : R^+ \rightarrow R, f(x) = \log_a (x)$, where $a >1$, and their graphs, as the inverse function of $y = a^x$ , including the relationships $a^{\log_a(x)} = x$ and $\log_a (a^x) = x$
• simple applications of exponential functions of the above form, with examples from various modelling contexts, and the interpretation of initial value, rate of growth or decay, half-life and long-run value in these contexts and their relationship to the parameters $A, k$ and $C$

source – VCE Mathematics Study Design

VCE Mathematical Methods Units 1 and 2 Courses

VCE Mathematical Methods Units 1 and 2 Syllabus

## Course Content

### Coordinate Geometry

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Functions
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Polynomials
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Trigonometry
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Exponential Graphs
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Logarithmic Graphs
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Exponential Growth and Decay
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• 65 Lessons
• 411 Topics
• 84 Quizzes