Polynomials, curve sketching and transformations flashcards

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Polynomial
A sum of terms axⁿ where every power n is a whole number ≥ 0; the degree is the highest power (a cubic has degree 3).
Factorise completely
Write as a product of factors none of which can be factorised further (check the quadratic left for a difference of two squares).
Asymptote
A line that a curve approaches more and more closely, without reaching it, as x or y grows without limit.
Repeated (double) root
A root from a squared factor (x − p)²: the curve touches the x-axis at x = p — a turning point on the axis — instead of crossing.
Translation
A transformation that slides every point of a graph the same distance in the same direction, without changing its shape.
Stretch parallel to the y-axis, scale factor a
Every y-coordinate is multiplied by a; x-coordinates (and so the roots) are unchanged.
Stretch parallel to the x-axis, scale factor 1/a
Every x-coordinate is divided by a; y-coordinates (and so the y-intercept) are unchanged.
Difference of two squares
x² − a² = (x − a)(x + a). (No — learn it)
y-intercept of y = (x − p)(x − q)(x − r)
Put x = 0: y = −pqr. (No — learn it)
y = f(x) + a
Translation a units in the positive y-direction: (x, y) → (x, y + a). (No — learn it)
y = f(x + a)
Translation a units in the NEGATIVE x-direction: (x, y) → (x − a, y). (No — learn it)
y = af(x)
Stretch parallel to the y-axis, scale factor a: (x, y) → (x, ay). (No — learn it)
y = f(ax)
Stretch parallel to the x-axis, scale factor 1/a: (x, y) → (x/a, y). (No — learn it)
Asymptotes of y = k/x and y = k/x²
x = 0 and y = 0 (the coordinate axes). (No — learn it)
Shape of a cubic with a positive x³ coefficient
Starts bottom left, ends top right; up to two turning points (a maximum then a minimum).
Shape of a cubic with a negative x³ coefficient
Starts top left, ends bottom right; up to two turning points (a minimum then a maximum).
y = k/x for k > 0 and for k < 0
k > 0: two branches in the 1st and 3rd quadrants. k < 0: 2nd and 4th quadrants. Never meets the axes.
y = k/x² for k > 0
Two branches, both above the x-axis, symmetrical about the y-axis; asymptotes x = 0 and y = 0.
y = (x − 1)³ at x = 1
The curve flattens and crosses the x-axis at (1, 0) — a triple root, not a turning point.
y = sin 2x compared with y = sin x
Stretch parallel to the x-axis, scale factor ½: period 180°, two full cycles in 0° ≤ x ≤ 360°.
Why does f(x + a) move the graph LEFT?
The new graph takes the old value f(p) when x + a = p, i.e. at x = p − a — a units to the left.
How do you find the number of solutions of f(x) = g(x) from sketches?
Count the points where y = f(x) and y = g(x) meet; each one gives one real solution.
Why can y = 3/x² never be negative?
x² > 0 for every x ≠ 0, so 3/x² has the sign of 3.
What happens to asymptotes under a transformation?
They move with the graph: f(x + a) moves x = c to x = c − a; f(x) + a moves y = d to y = d + a.
When does y = f(x) + k touch the x-axis?
When the translation puts a turning point exactly on the axis: k = minus the y-coordinate of that turning point.
Which points stay fixed under y = af(x)?
Points on the x-axis (y = 0), because a × 0 = 0 — the roots do not move.
Expand (x + 3)(x − 1)(x − 2).
(x − 1)(x − 2) = x² − 3x + 2; × (x + 3) gives x³ − 7x + 6.
Factorise completely 2x³ − 18x.
2x(x² − 9) = 2x(x − 3)(x + 3).
Key points of y = (x + 2)(x − 5)².
Positive cubic; crosses at (−2, 0), touches at (5, 0); y-intercept 2 × 25 = 50, so (0, 50).
y = f(x) has a minimum at (4, 7). Find the minimum of f(x + 3), f(2x) and 3f(x).
f(x + 3): (1, 7). f(2x): (2, 7). 3f(x): (4, 21).
Asymptotes and axis crossing of y = 2/(x − 1).
x = 1 and y = 0; x = 0 gives y = −2, so (0, −2). It never meets the x-axis.
Asymptotes and x-intercepts of y = 1/x² − 4.
x = 0 and y = −4; 1/x² = 4 gives x = ±½.
How many times does y = x³ meet y = −1/x?
Never: the two curves always have opposite signs (x⁴ = −1 has no real solution).
Zeros of y = cos(x − 90°) for 0 ≤ x ≤ 360°.
Translation 90° to the right: x − 90° = −90°, 90°, 270° gives x = 0°, 180°, 360° (it is the graph of sin x).
Sketch with no coordinates labelled
Label every axis crossing with its coordinates and every asymptote with its equation, on the sketch itself.
f(2x): doubling the x-coordinates
Wrong way round — f(2x) halves x-coordinates; f(½x) doubles them.
Dividing x³ − 5x² + 6x = 0 by x
Loses the root x = 0. Factor out x instead: x(x − 2)(x − 3) = 0.
“Hence state the number of solutions”
Show the algebra that turns the equation into “curve 1 = curve 2” (x ≠ 0), then count intersections and say so in words.

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