Flashcards · 7357 · Topic 3

B: Algebra and functions flashcards

56 cards on Topic 3, subtopics B1 to B11. Pick a subtopic or study them all.

Study mode needs JavaScript. Every card is listed below, so you can still cover the answers and test yourself.

All 56 cards

For a quick read-through or printing. Cover the right-hand side to test yourself.

B1 Laws of indices for rational exponents

The core laws of indices: aᵐ × aⁿ = aᵐ⁺ⁿ aᵐ ÷ aⁿ = aᵐ⁻ⁿ (aᵐ)ⁿ = aᵐⁿ a⁰ = 1 (a ≠ 0) Same base only.
What do a⁻ⁿ and a^(m/n) mean?: a⁻ⁿ = 1/aⁿ (a reciprocal, not a negative number). a^(m/n) = (ⁿ√a)ᵐ = ⁿ√(aᵐ).
Evaluate 27^(−2/3).: Negative index: reciprocal. Cube root of 27 is 3; squared is 9. 27^(−2/3) = 1/9.
Solve 2^(x + 1) = 8ˣ.: Write both sides as powers of 2: 2^(x + 1) = 2^(3x), so x + 1 = 3x and x = ½.
Simplify (2x³)⁴.: Apply the power to every factor, including the number: 2⁴x¹² = 16x¹².

B2 Surds

Simplify √75 and 3√8 + √18.: √75 = √25 × √3 = 5√3. 3√8 + √18 = 6√2 + 3√2 = 9√2.
How do you rationalise k/√a?: Multiply top and bottom by √a: k/√a = k√a/a.
Rationalise 4/(3 + √5).: Multiply top and bottom by the conjugate 3 − √5: 4(3 − √5)/(9 − 5) = 3 − √5.
Why does multiplying by the conjugate work?: (a + √b)(a − √b) = a² − b, which has no surd, so the denominator becomes rational.
Is √a + √b equal to √(a + b)?: No. √(ab) = √a × √b, but there is no such rule for addition: √9 + √16 = 7, while √25 = 5.

B3 Quadratic functions and their graphs

What does each sign of the discriminant b² − 4ac tell you?: > 0: two distinct real roots. = 0: one repeated (equal) root. < 0: no real roots.
Completed-square form a(x + p)² + q: what does it reveal?: The turning point (−p, q) and the line of symmetry x = −p. q is the minimum value if a > 0, the maximum if a < 0.
Write x² − 6x + 11 in completed-square form. Why is it always positive?: (x − 3)² − 9 + 11 = (x − 3)² + 2. (x − 3)² ≥ 0, so the expression is at least 2. Minimum point (3, 2).
The quadratic formula: x = (−b ± √(b² − 4ac))/(2a) for ax² + bx + c = 0. It works for any quadratic; factorising is quicker only when it is clean.
Solve x⁴ − 5x² + 4 = 0.: Let u = x²: u² − 5u + 4 = 0, so u = 1 or u = 4. x² = 1 or x² = 4, so x = ±1 or x = ±2.

B4 Simultaneous equations

Solving two linear simultaneous equations by elimination: 1. Scale one or both equations so one letter has equal coefficients. 2. Add if the signs differ, subtract if they match. 3. Solve, then substitute back. 4. Check in the equation you did not use.
Solving one linear and one quadratic equation by substitution: 1. Rearrange the linear equation for one letter. 2. Substitute into the quadratic, with brackets. 3. Solve the quadratic. 4. Use the linear equation to find each matching value; give answers as pairs.
After substituting a line into a quadratic curve, what does the discriminant show?: > 0: the line meets the curve twice. = 0: the line is a tangent (touches once). < 0: the line misses the curve.
Solve y = x + 1 and y = x² − 5.: x² − 5 = x + 1, so x² − x − 6 = 0 and (x − 3)(x + 2) = 0. Solutions (3, 4) and (−2, −1).
What can happen when you solve two linear simultaneous equations?: One solution (the lines meet once), no solution (parallel lines), or infinitely many (the same line).

B5 Linear and quadratic inequalities

When must you reverse an inequality sign?: When you multiply or divide both sides by a negative number. Do not multiply by an expression in x unless you know its sign.
Solve x² − x − 6 > 0.: Roots: (x − 3)(x + 2) = 0, x = −2 or 3. The ∪-shaped curve is above the axis outside the roots. x < −2 or x > 3.
Write "x < −2 or x > 3" and "−2 < x < 3" in set notation.: {x : x < −2} ∪ {x : x > 3} {x : −2 < x < 3} ∪ means "or"; ∩ means "and".
Shading a region for an inequality on a graph: strict and non-strict boundaries: Strict (< or >): dashed boundary line. Non-strict (≤ or ≥): solid boundary line.
Why sketch the graph before writing the solution of a quadratic inequality?: It shows whether the solution is between the roots (one interval, "and") or outside them (two intervals, "or"), which prevents writing it the wrong way round.

B6 Manipulating polynomials

Factor theorem: f(a) = 0 exactly when (x − a) is a factor of f(x).
Remainder when f(x) is divided by (x − a): f(a). The identity is f(x) ≡ (x − a) × quotient + remainder.
Show that (x − 2) is a factor of x³ − 3x² + 4 and factorise fully.: f(2) = 8 − 12 + 4 = 0, so (x − 2) is a factor. x³ − 3x² + 4 = (x − 2)(x² − x − 2) = (x − 2)²(x + 1).
Expand (x + 2)(x − 4)(3x + 1).: (x² − 2x − 8)(3x + 1) = 3x³ − 5x² − 26x − 8. Check at x = 1: 3 × (−3) × 4 = −36 and 3 − 5 − 26 − 8 = −36.
Simplify (x² − 9)/(x² + x − 6).: (x − 3)(x + 3)/((x + 3)(x − 2)) = (x − 3)/(x − 2), for x ≠ −3 (and x ≠ 2). Cancel whole factors only.
How do you factorise a cubic with no obvious common factor?: 1. Test factors of the constant term in f(x) until f(r) = 0. 2. Write (x − r) as a factor. 3. Divide (or compare coefficients) to get a quadratic. 4. Factorise the quadratic and state all roots.

B7 Graphs of functions

How does a polynomial graph behave at a single root, a double root (x − a)² and a triple root (x − a)³?: Single root: crosses the x-axis. Double root: touches the axis and turns back. Triple root: crosses with a flat (stationary) point.
Sketch y = |2x − 1|: vertex, y-intercept and arm gradients: Vertex at (1/2, 0), y-intercept (0, 1), arm gradients +2 and −2. The part of y = 2x − 1 below the x-axis is reflected above it.
What does y ∝ x mean, and y ∝ 1/x?: y ∝ x: y = kx for a constant k. y ∝ 1/x: y = k/x. Find k from one pair of values.
Asymptotes and shape of y = a/x and y = a/x² (a > 0): Both: asymptotes x = 0 and y = 0. y = a/x: first and third quadrants. y = a/x²: above the x-axis on both sides.
The ends of a cubic graph: Positive x³ coefficient: from bottom left to top right. Negative x³ coefficient: from top left to bottom right.

B8 Composite and inverse functions

What does gf(x) mean?: g(f(x)): apply f first, then g. Usually gf(x) ≠ fg(x).
f(x) = 2x + 1 and g(x) = x². Find fg(x) and gf(x).: fg(x) = 2x² + 1. gf(x) = (2x + 1)².
Which functions have an inverse?: Only one-one functions (each output comes from only one input). f⁻¹ undoes f: f⁻¹f(x) = x and ff⁻¹(x) = x.
Domain, range and graph of f⁻¹: Domain of f⁻¹ = range of f; range of f⁻¹ = domain of f. The graph of y = f⁻¹(x) is the reflection of y = f(x) in y = x.
Find f⁻¹(x) for f(x) = (2x + 3)/(x − 1), x ≠ 1.: y(x − 1) = 2x + 3, so x(y − 2) = y + 3 and x = (y + 3)/(y − 2). f⁻¹(x) = (x + 3)/(x − 2), x ≠ 2.

B9 Transformations of graphs

Describe y = f(x) + a, y = f(x + a), y = af(x) and y = f(ax).: f(x) + a: translation by (0, a). f(x + a): translation by (−a, 0). af(x): stretch parallel to the y-axis, scale factor a. f(ax): stretch parallel to the x-axis, scale factor 1/a.
Describe y = −f(x) and y = f(−x).: y = −f(x): reflection in the x-axis. y = f(−x): reflection in the y-axis.
Changes inside and outside the bracket of f: Outside: acts on y and does the obvious thing. Inside: acts on x and does the opposite (f(x + 3) moves left 3; f(2x) halves the x-values).
(2, 5) lies on y = f(x). Find its image on y = 2f(x − 3).: x-coordinate + 3; y-coordinate × 2. The image is (5, 10).
Which features stay fixed under a stretch parallel to the y-axis, and under a stretch parallel to the x-axis?: Stretch parallel to the y-axis: the x-intercepts stay fixed. Stretch parallel to the x-axis: the y-intercept stays fixed.

B10 Partial fractions

Partial fractions: the forms for (ax + b)(cx + d), (ax + b)² and (ax + b)(cx + d)²: A/(ax + b) + B/(cx + d) A/(ax + b) + B/(ax + b)² A/(ax + b) + B/(cx + d) + C/(cx + d)² Each numerator is a constant.
Express (9x − 5)/((3x − 1)(x − 1)) in partial fractions.: 9x − 5 ≡ A(x − 1) + B(3x − 1). x = 1: 4 = 2B, B = 2. x = 1/3: −2 = −(2/3)A, A = 3. 3/(3x − 1) + 2/(x − 1).
How do you find the constants in partial fractions?: 1. Multiply through by the denominator to get an identity. 2. Substitute values that make a factor zero. 3. Compare coefficients (or use another value) for any constant left. 4. Check with a value not yet used.
How many partial fractions should there be?: As many as the degree of the denominator, with one constant for each power of each factor.
What are partial fractions used for later in the course?: Integrating rational functions (each part gives a ln term) and expanding with the binomial series.

B11 Functions in modelling

Which type of function models a quantity that approaches a limit?: A − Be^(−kt) (k > 0), which has a horizontal asymptote at A.
Which function models a repeating pattern, and what do its parameters show?: a + b sin(kt) or a + b cos(kt): midline a, amplitude b, period 2π/k (360°/k in degrees).
What should you check about the domain of a model?: Which inputs make sense: watch for negative values, values beyond a physical maximum, and division by zero.
Warning signs in the long-term behaviour of a model: Unbounded growth, negative values of a quantity that cannot be negative, and exact repeats in real data.
How do you refine a model?: Name a specific change: add a limit, add a term, use a piecewise rule, or gather more data, then test it again.