Year 12: Functions

4 dot points across 4 inquiry questions, in syllabus order. Each dot point has a focused answer, with exam-style questions and worked answers where available.

What are the key features of exponential and logarithmic graphs, and how do transformations and the inverse relationship link them?

Sketch and interpret graphs of exponential and logarithmic functions, including transformations, and use the inverse relationship between them

Exponential and logarithmic graphs. Key features of $e^x$ and $\ln x$ , their inverse relationship, transformations, asymptotes, and graphs of related forms such as $e^{-x}$ and $\ln(x + a)$ , with worked examples.

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How do we sketch graphs built from sums, differences, products, quotients and reciprocals of standard functions?

Sketch graphs of sums, differences, products, quotients, squares and reciprocals of two known functions

Combining functions graphically. How to build sketches of $f + g$ , $f g$ , $f / g$ , $1 / f$ and $f^2$ from the graphs of $f$ and $g$ , where features come from, and what asymptotes and zeros do, with worked examples.

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When does a function have an inverse, and how do we form, evaluate and graph composite and inverse functions?

Form composite functions, determine when a function has an inverse, find and graph the inverse, and use restriction of domain to invert non-one-to-one functions

Composite and inverse functions. Composition order and domain, the horizontal line test, finding the inverse by swapping and solving, the reflection in $y = x$ , and restricting domains to invert non-one-to-one functions, with worked examples.

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How do translations, reflections and dilations transform the graph of a function in a predictable way?

Apply translations, reflections and dilations to the graph of a function and identify the resulting equation

Graph transformations. Vertical and horizontal translations, reflections in the axes, vertical and horizontal dilations, the order of combined transformations, and how each affects the equation, with worked examples.

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