Math Analysis & Approaches unit guide
Functions
Graphs, transformations, rational and polynomial functions, inequalities and the modulus.
Functions is the unit that makes the rest of the course readable. Every calculus question is a question about a function, and students who can picture a graph from its equation find differentiation and integration far less abstract than those who cannot. The single most valuable habit here is sketching before calculating, even roughly.
Topics in this unit
What gets examined
- Domain and range, and stating them correctly with the right notation.
- Composite and inverse functions, including the effect of composition on domain.
- Transformations of graphs, and the order in which combined transformations apply.
- Asymptotes of rational functions, both vertical and horizontal, and what happens between them.
- Solving polynomial equations using the factor and remainder theorems.
- Inequalities, including where multiplying by a negative or an unknown changes the direction.
How to revise it
Sketch first, always
Intercepts, asymptotes, turning points. A rough sketch catches the sign errors and missing solutions that pure algebra hides, and sketching is examined directly as well.
Get the order of transformations right
Inside the bracket affects x and does the opposite of what it looks like; outside affects y and behaves as expected. Combined transformations apply in a specific order, and reversing it is a standard trap.
Find asymptotes before plotting anything
Vertical where the denominator is zero, horizontal from the behaviour as x grows. They frame the whole sketch, so finding them first makes the rest fall into place.
Solve inequalities by region, not by rearranging
Find the critical values, then test each interval. Multiplying through by an expression that might be negative is where most inequality marks are lost.
Where marks get lost
- Giving the range when the domain was asked for.
- Applying a horizontal stretch in the wrong direction.
- Cancelling a factor in a rational function and losing the hole it leaves behind.
- Multiplying an inequality by a variable without considering its sign.
- Assuming an inverse exists without checking the function is one-to-one.
Practise this unit
Reading about a unit only goes so far. Work through real IB-style questions on it, with mark schemes.