Mathematics Advanced - Hard Topic Revision

Vocabulary revision cards for Mathematics Advanced covering Preliminary topics including quadratics, calculus, trigonometry, logarithms, exponentials, and functions.

Discriminant

Defined by $Δ = b^2 - 4ac$ for a quadratic equation; determines root nature where $Δ > 0$ gives $2$ real roots, $Δ = 0$ gives $1$ repeated real root, and $Δ < 0$ gives no real roots.

Sum and Product of Quadratic Roots

For $ax^2 + bx + c = 0$ with roots ̑\alpha and ̑\beta, the sum of roots is ̑\alpha + ̑\beta = -\frac{b}{a} and the product of roots is ̑\alpha\beta = \frac{c}{a}.

Difference of Roots Squared Formula

Given by $(\alpha - \beta)^2 = (\alpha + \beta)^2 - 4\alpha\beta$.

Absolute Value Equations

Equations where $|x| = a$ resolves to $x = a$ OR $x = -a$, and $|A| = |B|$ resolves to $A = B$ OR $A = -B$.

Absolute Value Inequalities

Inequalities where $|x| < a$ simplifies to $-a < x < a$, and $|x| > a$ simplifies to $x < -a$ OR $x > a$.

Domain

All allowed $x$-values for a function.

Square Root Domain Rule

For $f(x) = \sqrt{g(x)}$, requires $g(x) \ge 0$.

Fraction Domain Rule

For $f(x) = \frac{1}{g(x)}$, requires $g(x) \neq 0$.

Range Finding Method

Let $y = f(x)$, rearrange to make $x$ the subject, then determine which $y$-values are possible.

Vertical Asymptote

Found by setting the denominator equal to $0$ AFTER cancelling common factors in a rational function.

Horizontal Asymptote Rules

If numerator degree < denominator degree, $y = 0$; if degrees are equal, $y$ equals the ratio of leading coefficients.

Oblique Asymptote

An asymptote found using polynomial division when the numerator degree of a rational function is exactly $1$ greater than the denominator degree.

Function Transformation Inside vs Outside

Transformations outside $f(x)$ affect vertical movement, while transformations inside $f(x)$ affect horizontal movement.

CAST Rule

Quadrant rule for sign determination of trigonometric functions where A = All positive, S = Sine positive, T = Tan positive, and C = Cosine positive.

Pythagorean Trigonometric Identity

The identity ̑\sin^2(x) + \cos^2(x) = 1.

Tangent and Secant Identities

Identity definitions given by $\tan(x) = \frac{\sin(x)}{\cos(x)}$ and $\sec(x) = \frac{1}{\cos(x)}$.

Logarithm Argument Constraint

The rule that every logarithm argument must be POSITIVE.

Logarithm Product Rule

$$\log_a(xy) = \log_a(x) + \log_a(y)$$

Logarithm Quotient Rule

$$\log_a\left(\frac{x}{y}\right) = \log_a(x) - \log_a(y)$$

Logarithm Power Rule

$$\log_a(x^n) = n\log_a(x)$$

Exponential Quadratic Substitution

A technique where expressions with $e^{2x}$ and $e^x$ are solved by substituting $u = e^x$, turning $e^{2x}$ into $u^2$.

Product Rule

Differentiation rule stating that if $y = uv$, then $y' = u'v + uv'$.

Quotient Rule

Differentiation rule stating that if $y = \frac{u}{v}$, then $y' = \frac{u'v - uv'}{v^2}$.

Chain Rule

Differentiation rule stating that if $y = [f(x)]^n$, then $\frac{dy}{dx} = n[f(x)]^{n-1} \times f'(x)$.

Stationary Point

A point on a curve where $\frac{dy}{dx} = 0$.

Second Derivative Test for Local Extrema

Method to classify stationary points where $f''(x) > 0$ indicates a minimum and $f''(x) < 0$ indicates a maximum.

Tangent Line Gradient

Given by $m = f'(x)$ at a specified point.

Normal Line Gradient

Given by $m_{\text{normal}} = -\frac{1}{m_{\text{tangent}}}$.

"Hence" Keyword

An exam directive indicating to use the previous result to solve the section.

"Exact" Keyword

An exam requirement specifying that the answer must be given without decimals.