Unit 2: Algebra, Relations and Functions — The Quadratic Function, Polynomial and Rational Functions and Their Operations

• A quadratic function is f(x) = ax² + bx + c with a, b, c real and a ≠ 0; its graph is a parabola.
• The function value at x = k is f(k): substitute and compute — a point (k, m) lies on the curve exactly when f(k) = m.
• To find an unknown coefficient from a point on the curve, substitute the point and solve the resulting linear equation.
• The y-intercept is (0, c); the x-intercepts are the zeros of f.

• In vertex form the vertex is (h, k) and the axis of symmetry is the line x = h.
• a > 0 the parabola opens upward and k is the minimum value; a < 0 it opens downward and k is the maximum value.
• Watch the sign inside the bracket: y = (x − 3)² + 2 has vertex (3, 2), while y = (x + 3)² − 2 has vertex (−3, −2).

Equation from the vertex and one point

• Write y = a(x − h)² + k with the given vertex, then substitute the other point to obtain a.
• Vertex (1, −4) through (3, 0): 0 = a(3 − 1)² − 4 → 4a = 4 → a = 1, so y = (x − 1)² − 4 = x² − 2x − 3.

• For f(x) = ax² + bx + c: h = −b/(2a) and k = f(h) = (4ac − b²)/(4a).
• Completing the square converts between the two forms: x² + 6x + 5 = (x + 3)² − 4, vertex (−3, −4).
• Zeros from the general form: solve ax² + bx + c = 0; the axis x = h is exactly halfway between the two zeros.

• Plot, in order: the vertex, the y-intercept (0, c), the reflection of the y-intercept across the axis (2h, c), and the x-intercepts when they exist.
• Δ = b² − 4ac tells how many x-intercepts: two if Δ > 0, one (the vertex touches the axis) if Δ = 0, none if Δ < 0.
• Draw a smooth symmetric curve, never a "V" — the parabola is curved everywhere.
• Translations: y = a(x − h)² + k is the image of y = ax² shifted h right and k up.

• A quantity modelled by a quadratic reaches its extremum at the vertex, i.e. at x = −b/(2a); the extremum value is k.
• Fenced area: with a fixed perimeter the area A(x) is quadratic in x and maximal at the vertex — the answer is always a square.
• Profit: revenue minus cost is quadratic; the maximum profit is the vertex value, and break-even points are its zeros.
• Projectile height: h(t) = −16t² + v₀t + h₀ opens downward, so the vertex gives the greatest height and its time.
• Reject any value of the variable that the situation forbids (negative time, negative price).

• A polynomial function is f(x) = aₙxⁿ + … + a₁x + a₀ with aₙ ≠ 0; n is the degree and aₙ the leading coefficient.
• Names by degree: 1 linear, 2 quadratic, 3 cubic; constants have degree 0.
• The end behaviour follows the leading term: even degree with a positive coefficient rises on both sides; odd degree with a positive coefficient falls to the left and rises to the right.

By factoring

• Set f(x) = 0 and factor completely; each factor gives a zero. x³ − 4x = x(x − 2)(x + 2) → zeros 0, 2, −2.

By completing the square

• x² − 6x + 4 = 0 → (x − 3)² = 5 → x = 3 ± √5.

From the graph

• A zero is an x-intercept. If the curve crosses the axis the multiplicity is odd; if it only touches and turns the multiplicity is even.
• A polynomial of degree n has at most n real zeros, and the sum of the multiplicities equals n.

• A rational function is f(x) = P(x)/Q(x), Q ≠ 0, where P and Q are polynomials.
• Domain: all reals except the values that make Q(x) = 0. Example: Q(x) = x² − 9 excludes x = ±3, so the domain is ℝ − {−3, 3}.
• Simplifying: factor numerator and denominator, cancel only common factors, and keep the excluded values as restrictions. (x² − 1)/(x² + x) = (x − 1)(x + 1)/[x(x + 1)] = (x − 1)/x, x ≠ 0, ±1… restrictions x ≠ 0, x ≠ −1.
• Zeros of a rational function: the zeros of the numerator that belong to the domain — a value that also vanishes the denominator is not a zero.

• The common domain of two or more functions is the intersection of their domains; any operation between them is defined only there.
• Two functions are equal when they have the same domain and the same value for every element of it — identical simplified formulas on different domains are not the same function.
• When asked for the domain of a sum, product or quotient, start from the intersection and then remove the zeros of any denominator that appears.

• Addition and subtraction: use the lowest common denominator, rewrite each fraction over it, then add or subtract the numerators only, keeping the common denominator.
• Multiplication: factor everything, cancel common factors across the numerator and denominator, then multiply straight across.
• Division: multiply by the reciprocal of the divisor, and add to the restrictions the zeros of the new numerator that came from flipping.
• State the final answer in simplest form together with its domain restrictions.

• The additive inverse of f is −f; their sum is the zero function, the additive identity.
• The multiplicative inverse (reciprocal) of f = P/Q is Q/P, defined only where P ≠ 0 and Q ≠ 0; their product is 1.
• Properties on the common domain: closure, commutativity and associativity of addition and multiplication, the distributive law of multiplication over addition, and the identities 0 (additive) and 1 (multiplicative).
• Subtraction and division are not commutative and not associative — do not extend those properties to them.