Flashcards · 4MA1 · Topic 3

Sequences, functions and graphs flashcards

26 cards on Topic 3, subtopics 3.1 to 3.4. Pick a subtopic or study them all.

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Term-to-term rule: How to get the next term from the one before, for example "add 4" or "multiply by 2".
Position-to-term rule: A formula for the nth term in terms of n, for example 7n − 4. It gives any term directly.
Find the nth term of 13, 9, 5, 1, …: Difference −4, so start with −4n: −4, −8, −12, … Each term is 17 more: nth term = 17 − 4n.
How do you test whether a number is a term of a sequence?: Set the nth term equal to the number and solve for n. A whole-number n means it is a term; otherwise it is not.
nth term of an arithmetic sequence with first term a and common difference dHigher: a + (n − 1)d. Learn it: it is not on the formulae sheet.
Find the sum of the first 15 terms of 2 + 5 + 8 + 11 + …Higher: Sₙ = (n/2)[2a + (n − 1)d], with a = 2, d = 3. S₁₅ = (15/2)(4 + 14 × 3) = (15/2)(46) = 345

FunctionHigher: A mapping that sends each input to exactly one output. Written f(x) = … or f : x ↦ …
Domain and range of a functionHigher: Domain: the allowed inputs. Range: the resulting outputs.
f(x) = x + 2 and g(x) = 3x. Find fg(x) and gf(x).Higher: fg(x): do g first, then f: 3x + 2. gf(x): do f first, then g: 3(x + 2) = 3x + 6.
f(x) = (x − 7)/4. Find f⁻¹(x).Higher: y = (x − 7)/4, so x = 4y + 7. Replace y with x: f⁻¹(x) = 4x + 7.
Is f⁻¹(x) the same as 1/f(x)?Higher: No. f⁻¹ is the inverse function, which reverses f. 1/f(x) is the reciprocal.
Which value of x must be excluded from the domain of f(x) = 1/(x − 3)?Higher: x = 3, the value that makes the denominator zero (not the value of the denominator itself).

Gradient of a line: (Increase in y) ÷ (increase in x). Example: through (−2, 5) and (4, −7): −12 ÷ 6 = −2.
Shape of a quadratic graph: U-shape if the x² coefficient is positive, ∩-shape if negative; symmetrical about a vertical line through the turning point.
Distance–time graphs: what do steep, horizontal and downward lines show?: Steeper line: faster. Horizontal line: stopped. Line sloping back down: returning to the start.
Find the equation of the line parallel to y = 3x − 1 through (2, 9)Higher: Parallel, so gradient 3. Substitute: 9 = 3 × 2 + c, so c = 3. y = 3x + 3
Gradient of a line perpendicular to a line of gradient mHigher: −1/m (the negative reciprocal), because m₁ × m₂ = −1. Example: perpendicular to gradient −1/4 is gradient 4.
Shapes of cubic and reciprocal graphsHigher: Cubic: S-shaped; it ends top right if the x³ coefficient is positive, bottom right if negative. y = k/x: two branches that never meet the axes.
Trigonometric graphs: period and rangeHigher: sin and cos repeat every 360° and stay between −1 and 1. tan repeats every 180°, with asymptotes at 90°, 270°, …
How do y = f(x) + a and y = f(x + a) transform the graph of y = f(x)?Higher: y = f(x) + a: translation up a; (p, q) → (p, q + a). y = f(x + a): translation left a; (p, q) → (p − a, q).

Differentiate y = axⁿHigher: dy/dx = naxⁿ⁻¹: multiply by the power, then reduce the power by 1.
Differentiate y = 5x⁴ − 3/xHigher: Write 3/x as 3x⁻¹: y = 5x⁴ − 3x⁻¹. dy/dx = 20x³ + 3x⁻² = 20x³ + 3/x²
Stationary (turning) pointHigher: A point where dy/dx = 0.
Find the turning point of y = x² − 6x + 1Higher: dy/dx = 2x − 6 = 0, so x = 3. y = 9 − 18 + 1 = −8 (substitute into the original equation). (3, −8) is a minimum (U-shaped graph).
Kinematics: how are displacement s, velocity v and acceleration a linked?Higher: v = ds/dt (differentiate displacement). a = dv/dt (differentiate velocity).
"At rest" and "maximum velocity": what condition does each give?Higher: At rest: v = 0. Maximum velocity: a = 0.