MB Pre-Calculus 40S common mistakes
MB Pre-Calculus 40S · Understand the traps and how to avoid them.
When applying a transformation of the form y = af(b(x - h)) + k, the candidate reads the horizontal shift directly from h and moves the graph in the direction of the sign (right for h > 0) instead of the opposite direction.
The candidate treats the horizontal parameter like the vertical one, forgetting that the shift is embedded inside the function argument, so the graph moves opposite to the sign that appears in the expression after factoring b out of the argument.
What to do instead
Always factor the coefficient of x out of the argument first, so the expression reads b(x - h). Then state the shift as h units in the direction opposite to the sign shown, and verify by testing one point: find the input that makes the argument zero and confirm that input lands at the new horizontal position.
The candidate transforms points of a parent function by applying the stretch and the translation in whatever order they read the expression, producing points that do not lie on the final graph.
The candidate does not distinguish between the mapping of points (which applies stretches and reflections relative to the axes before translations) and the order of operations inside the equation.
What to do instead
Use the mapping rule explicitly: for y = af(b(x - h)) + k, map each parent point (x, y) to (x/b + h, ay + k). Practise by transforming three key points of the parent graph, plotting them, and checking that the resulting curve has the expected shape and orientation.
The candidate solves a radical equation by squaring both sides and reports every solution obtained, including values that make the original radicand negative or that fail the original equation.
Squaring is not reversible, so it can introduce extraneous roots, and the candidate treats the squared equation as equivalent to the original rather than as a one-way implication.
What to do instead
After solving, substitute every candidate back into the original equation and discard any that fail. Also record the domain restriction from each radicand before solving, and check candidates against that restriction as a second filter.
The candidate writes the domain and the vertical asymptote of a rational function using the zeros of the numerator instead of the zeros of the denominator, or cancels a common factor and then still reports the cancelled value as a vertical asymptote.
The candidate memorises 'set the bottom to zero' but confuses numerator and denominator roles, and does not realise that a cancelled factor produces a hole rather than an asymptote.
What to do instead
Factor numerator and denominator completely. List denominator zeros as excluded values; after cancelling, classify each excluded value as a vertical asymptote (factor remains in the denominator) or a hole (factor cancelled), and state the hole's y-coordinate by substituting into the simplified expression.
The candidate evaluates inverse trigonometric expressions such as arcsin(sin(5π/6)) by cancelling the functions and returning 5π/6, ignoring the restricted range of the inverse.
The candidate treats inverse functions as exact undo operations without accounting for the restricted domain and range chosen to make them functions.
What to do instead
Before evaluating, rewrite the inner angle into the principal range of the inverse function (for arcsine, between -π/2 and π/2; for arccosine, between 0 and π). Use reference angles and quadrant signs to convert, then evaluate. Drill with angles outside the principal range.
The candidate solves a trigonometric equation over a stated interval but reports only the solution from the inverse function, omitting the second solution in the same period or solutions in other periods within the interval.
The candidate forgets that sine and cosine take each value twice per period and that the general solution repeats every period, so a single principal value feels complete.
What to do instead
After finding the reference angle, use the CAST/quadrant rule to generate all solutions in one period, then add integer multiples of the period and filter to the required interval. For tangent, add multiples of π only. Always count expected solutions against the interval length.
The candidate applies logarithm laws to expressions such as log(a + b) or log(a)/log(b), rewriting them as log a + log b or log(a - b).
The candidate overgeneralises the product, quotient, and power laws into operations the laws do not cover, because the laws look like ordinary algebraic distributivity.
What to do instead
Write the three valid laws side by side with a list of invalid moves, and for each problem ask whether the log's argument is a product, quotient, or power before applying anything. When solving, expand or condense only after factoring the argument, and verify a numerical case with easy numbers.
The candidate solves an exponential equation by taking the logarithm of only one side, or drops the logarithm when isolating the exponent, producing an answer that is not a solution.
The candidate treats 'take the log' as an operation applied to a term rather than to the whole equation, and loses track of the equality while manipulating.
What to do instead
Isolate the exponential expression first, then take the logarithm of both sides as a whole equation, and use the power law to bring the exponent down. Check by substituting the answer back, or by estimating with nearby powers of the base.
The candidate uses the permutation formula for a selection problem where order does not matter, or uses combinations when the objects are being arranged into distinct positions, producing answers that are too large or too small by a factorial factor.
The candidate keys on words like 'choose' or 'arrange' without deciding whether different orderings of the same set count as different outcomes, and does not check whether repetition or identical items are involved.
What to do instead
Before computing, write one sample outcome and ask whether reordering it gives a different outcome. If yes, use permutations; if no, use combinations. Then check for restrictions (identical items, mandatory members, adjacency) and divide by factorials for repeats. Compare the magnitude of the answer with a small hand-counted case.
The candidate expands a binomial power and miscomputes a middle term by using the wrong exponent pairing, or writes the general term as C(n, r)a^r b^(n-r) with the exponents swapped relative to the intended term number.
The candidate memorises the general term loosely and does not anchor r to a clear convention, so off-by-one errors shift every coefficient and exponent.
What to do instead
Fix a convention and write it down each time: the term containing a^(n-r)b^r has coefficient C(n, r), and it is the (r + 1)th term. When a specific term is requested, solve for r first, then substitute. Verify by checking that the exponents in every term sum to n and that the first and last coefficients are 1.
The candidate identifies a conic from its equation by the presence of squared terms alone, mislabelling a circle as an ellipse or an ellipse as a hyperbola, and reads the centre from the wrong signs after completing the square.
The candidate relies on the general shape of the equation rather than comparing coefficients and signs, and makes sign errors when converting general form to standard form.
What to do instead
Complete the square into standard form first. Then compare: equal coefficients with the same sign means a circle; unequal positive coefficients means an ellipse; opposite signs means a hyperbola; only one squared term means a parabola. Read the centre as the values that make each squared binomial zero, and check against the original equation.
The candidate treats an arithmetic or geometric sequence as the corresponding series, or uses n instead of n - 1 in the general term, so the nth term and the sum of n terms are off by one term.
The candidate conflates the term formula with the sum formula and forgets that the first term corresponds to index 1, so the common difference or ratio is applied one time too few or too many.
What to do instead
Write both formulas explicitly and label them: t_n = a + (n - 1)d or t_n = ar^(n - 1), and S_n separately. For each problem, state whether a single term or a total is wanted before substituting. Verify with a short sequence written out by hand (first four terms) and confirm the formula reproduces them.
The candidate sums an infinite geometric series without checking that the common ratio satisfies |r| < 1, and reports a finite sum for a divergent series.
The candidate applies S = a/(1 - r) as a universal formula and does not connect convergence to the size of the ratio.
What to do instead
Compute r first and state the convergence condition |r| < 1 explicitly before using the infinite sum formula. If |r| is greater than or equal to 1, write that the series diverges and no finite sum exists. Include a quick check that the terms are shrinking toward zero.