Title: InkMaths / Algebra and functions
On every note of this deck, unless a note says otherwise:
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Front: RECALL
State the factor theorem.
Back: (x − p) is a factor of f(x) exactly when f(p) = 0. For a factor (ax − b) test f(b/a) instead, which is where most of the lost marks are. Showing f(p) = 0 without the sentence naming the factor usually costs the conclusion mark.

Front: RECALL
State which graph transformations go the way you do not expect.
Back: Anything inside the bracket acts on x and acts backwards, so y = f(x + a) moves left by a and, for a > 0, y = f(ax) squashes by a factor 1/a. Anything outside acts on y and behaves normally, so y = f(x) + a moves up and, for a > 0, y = af(x) stretches by a. A negative a scales by |a| and reflects as well, in the y-axis inside the bracket and in the x-axis outside it. Check with one point you can compute.

Front: RECALL
State the partial fraction form to write down.
Back: Each distinct linear factor gets a constant over it. A repeated factor needs two terms, A/(x − a) and B/(x − a)², and writing only the squared one leaves the identity unsolvable. If the numerator's degree reaches the denominator's, divide first and split the remainder.

Front: SPOT THE ERROR
Say what is wrong with this claim:
fg(x) means apply f first, then g.
Back: It means g first. The function nearest x acts first, because fg(x) is shorthand for f(g(x)). Test it on f(x) = x + 1 and g(x) = x². fg(3) = 10 and gf(3) = 16, so the order is worth six marks of difference.

Front: STANDARD RESULT — LEARN IT
Discriminant:
ax^2 + bx + c = 0: discriminant = ?\(ax^{2} + bx + c = 0: \text{discriminant} = ?\)ax^2 + bx + c = 0: discriminant = ?
Back: b^2 - 4ac. Positive gives two roots, zero gives one repeated root, negative gives none\(b^{2} - 4ac. \text{ Positive gives two roots, zero gives one repeated root, negative gives none}\)b^2 - 4ac. Positive gives two roots, zero gives one repeated root, negative gives none

Front: RECALL
State how to prove a line and a curve never meet.
Back: Substitute the line into the curve, gather everything on one side, and show the discriminant of the resulting quadratic is negative. Zero means the line is a tangent, and that is the same calculation with a different final sentence.

Front: RECALL
State what completing the square hands you.
Back: Write the quadratic as a(x + p)² + q. The turning point sits at (−p, q), the minimum value of the expression is q when a is positive, and if q has the same sign as a there are no real roots. Three marks from one rewrite, so do it before you reach for the formula.

Front: RECALL
State what an inverse function does to the domain and the range.
Back: They swap. The domain of f inverse is the range of f, and its range is the domain of f. The inverse exists only if f is one-to-one. That is what a restricted domain in the question is there for. The two graphs are reflections of each other in the line y = x.