Mathematics Methods - Foundation (MTM315117) Practice Flashcards

A set of vocabulary flashcards covering key algebra and function concepts from the Mathematics Methods - Foundation MTM315117 lecture notes and exams.

Discriminant ($\Delta$)

A value calculated by $b^2 - 4ac$ used to predict the number and type of solutions for a quadratic equation; for example, if $\Delta = 364$, there are two real irrational solutions.

Factor Theorem

A theorem stating that if a polynomial $P(x)$ satisfies $P(k) = 0$, then $(x - k)$ is a factor of that polynomial, such as $(x - 3)$ being a factor of $x^3 - 2x^2 - 5x + 6$.

Remainder Theorem

A rule established in the lecture where the remainder of a polynomial $P(x)$ divided by $(x - a)$ is equal to $P(a)$, used to evaluate unknowns like $k$ when dividing $8x^3 - 5kx + 5$ by $(x - 2)$.

Point of Inflection

The specific coordinate where the curvature of a cubic function changes sign, such as $(1, 5)$ for the function $f(x) = -2(x - 1)^3 + 5$.

Turning Point

The vertex of a quadratic function representing its maximum or minimum value, such as the point $(4, -1)$ for the parabola $y = x^2 - 8x + 15$.

Parallel Lines

Lines that have the same gradient ($m_1 = m_2$), such as the line $y = \frac{1}{2}x + 1$ being parallel to $y = \frac{1}{2}x + 6$.

Perpendicular Lines

Lines whose gradients are negative reciprocals of each other ($m_1 \times m_2 = -1$), such as a line with gradient $-2$ being perpendicular to a line with gradient $\frac{1}{2}$.

Vertical Line Test

A method used to determine if a relation is a function; if any vertical line intersects the graph more than once, it fails the test and is not a function.

Domain

The set of all possible input $x$-values for a function, which may be restricted, such as $x \in [-3, 2)$ for the function $f(x) = (x + 2)^2(x - 1)$.

Range

The set of all possible output $y$-values that a function can produce, such as $y \in (-14.3, 30]$ for a specific graphed relation.

Completing the Square

An algebraic method used to solve quadratic equations or convert them into turning point form $(x + h)^2 + k = 0$, used for equations like $x^2 + 8x - 2 = 0$.

Quadratic Formula

The formula given by $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ used to find the exact or decimal solutions of any quadratic equation.

Binomial Expansion

The process of expanding expressions in the form $(ax + b)^n$ using Pascal's Triangle or the Binomial Theorem, such as expanding $(2x - 1)^4$ into $16x^4 - 32x^3 + 24x^2 - 8x + 1$.

Gradient ($m$)

The measure of the steepness of a line, calculated between two points $(x_1, y_1)$ and $(x_2, y_2)$ using the formula $m = \frac{y_2 - y_1}{x_2 - x_1}$.

Rational Solutions

Solutions to a quadratic equation that occur when the discriminant ($\Delta$) is a perfect square, as seen in the equation $5x^2 - 3x - 2 = 0$ where $\Delta = 49$.

Irrational Solutions

Solutions occurring when the discriminant is positive but not a perfect square, resulting in exact forms involving square roots, such as $\Delta = 160$.

No Real Solutions

A condition occurring when the discriminant is negative ($\Delta < 0$), indicating that a quadratic equation like $x^2 - x + (k + 3) = 0$ has no real roots when $k > -\frac{11}{4}$.

Simultaneous Equations

A set of two or more equations with common variables solved together to find a point of intersection, such as modeling pallet masses for washing powder ($WP$) and sugar ($S$).