Pre-Calculus

What this covers

• Analysing polynomial functions: multiplicity of zeros and what it does to the graph at each x-intercept, end behaviour from the leading term, and building a function from stated roots
• Rational functions: locating vertical asymptotes, holes, and x-intercepts from factored form, and determining horizontal or slant asymptotes by comparing degrees
• Transformations and composition: predicting the effect of f(2x - 6) + 1 on a graph, decomposing a complicated function into inner and outer parts, and finding domains of composite functions
• Inverse functions: testing invertibility, restricting a domain so an inverse exists, and reading the inverse relationship off a graph as reflection in y = x
• Conic sections: completing the square to get standard form, identifying centre, foci, vertices, and asymptotes of ellipses and hyperbolas
• Vectors, parametric equations and polar coordinates: component form, dot products, converting between rectangular and polar, and eliminating the parameter to recover a Cartesian equation

Where learners get stuck

Reading f⁻¹(x) as 1/f(x)

The notation borrows the exponent symbol, and students have spent years reading a superscript −1 as a reciprocal. It usually surfaces when a problem mixes both ideas, e.g. asking for f⁻¹(3) when f(3) is easy to compute. Working from the 'undo the machine' definition and checking with f(f⁻¹(x)) = x fixes it faster than restating the rule.

Assuming every zero of the denominator gives a vertical asymptote

Students learn 'denominator = 0 means asymptote' before they meet common factors. When a factor cancels, the graph has a removable hole instead, and the function is still undefined there. Insisting on full factorisation of numerator and denominator before drawing any conclusion is the habit that prevents this.

Applying horizontal transformations in the intuitive direction

y = f(x - 3) shifts right and y = f(3x) compresses, both of which feel backwards. The cause is that the transformation acts on the input before the function does, so the graph must compensate. Testing a single point — asking what x makes the inside equal a known value — makes the direction concrete rather than memorised.

What a session looks like

A session usually opens with one problem you are currently stuck on or a diagnostic question that reveals which sub-skill is shaky. You talk through your reasoning while working on paper or a graphing tool; Evelyn asks for the next step rather than supplying it, and pushes on the justification — why that asymptote, why that domain restriction. When an error appears, the underlying rule is retested on a fresh example before moving on. Sessions typically end with a short verbal recap of the method used and one problem to attempt independently.

Helpful to know first

• Confident factoring of quadratics, differences of squares, and simple cubics, plus polynomial long division or synthetic division
• Solving systems of linear equations and rational and radical equations, including checking for extraneous solutions
• Function notation, domain and range, and interpreting graphs of linear and quadratic functions
• Right-triangle trigonometry and the unit circle, which the polar and vector work draws on

Questions

Is Pre-Calculus the same as Algebra 2?

They overlap, but Pre-Calculus goes further. Algebra 2 introduces polynomial, rational and exponential functions; Pre-Calculus analyses their behaviour in detail and adds conic sections, vectors, parametric and polar representations, and complex numbers in trigonometric form. It also emphasises describing limits and end behaviour informally, which Algebra 2 generally does not.

How much trigonometry is in Pre-Calculus?

A substantial amount in most courses, but trigonometric identities, equations and graphs are handled as a separate topic here. This topic uses trigonometry where it supports polar coordinates, vector components and the polar form of complex numbers, and assumes the unit circle is already familiar.

My child gets the algebra right but loses marks on graphing questions. What helps?

Usually the gap is in translating algebraic facts into graph features and back — knowing that a factor of (x - 2)³ means the curve flattens and crosses at x = 2, or that degree of numerator minus degree of denominator equals one implies a slant asymptote. Sessions target that translation directly by asking for a sketch and its justification before any calculation is checked.

Does this prepare a student for a first calculus course?

It covers the algebraic groundwork calculus relies on: manipulating function forms fluently, composing and decomposing functions, and describing behaviour near undefined points. Whether that is enough for a specific calculus course depends on that course's syllabus, which can be reviewed at the start.