Mathematics: Linear and Quadratic Functions

A set of practice flashcards covering expressions, equations, variables, and the properties of linear and quadratic functions based on lecture notes.

What is a 'term' in a mathematical expression?

A term is each individual component of an expression, for example, $2x$ or $3y$ in the expression $2x + 3y$.

How are terms categorized based on their symbols?

Terms with the same symbols are called 'like terms,' while terms with different symbols are called 'unlike terms.'

What is formed when two mathematical expressions are equal?

An equation is formed when two expressions are equal; finding the value of the unknown that equates them is known as solving the equation.

What is a variable in mathematics?

A variable is a measurement of a phenomenon that can take different values, such as the weight of patients at a hospital.

How do dependent and independent variables differ?

A dependent variable is a measurement that depends on another variable (e.g., student marks), whereas an independent variable is a measurement that determines another (e.g., hours of study).

What is the definition of a function?

A function is an expression that shows how two or more variables are related, represented as $y = f(x)$, meaning $y$ is expressed in terms of $x$.

What defines a linear function?

A function is linear if the highest power of the independent variable is $1$, typically expressed in the form $y = f(x) = mx + c$.

In the linear function $y = mx + c$, what do the symbols $m$ and $c$ represent?

$m$ represents the gradient (slope) of the line, and $c$ represents the $y$ intercept where the line crosses the y-axis.

What is the general form of a quadratic function?

A quadratic function is expressed in the form $y = f(x) = ax^2 + bx + c$, where the highest power of the independent variable is $2$.

What is factorization in the context of functions?

Factorization is the process of finding factors, which are a set of expressions or numbers whose product results in the original function.

What are the steps for factorizing a quadratic function?

1. Find 2 numbers (factors) such that their product is $ac$ and their sum is $b$. 2. Replace the middle term with these numbers. 3. Factorize completely. 4. Write down the factors.

How does the sign of the gradient ($m$) affect the graph of a linear function?

If $m$ is positive, the graph is an upward sloping straight line; if $m$ is negative, the graph is a downwards sloping straight line.

What is the relationship between the gradients of parallel and perpendicular lines?

Parallel lines have the same gradient. For perpendicular lines, if one gradient is $m$, the other is given by $-\frac{1}{m}$.

How is the gradient ($m$) calculated when given two points $(x_1, y_1)$ and $(x_2, y_2)$?

It is calculated using the formula $m = \frac{y_2 - y_1}{x_2 - x_1}$, which represents the change in $y$ over the change in $x$.

How does the sign of $a$ determine the shape of a quadratic graph?

If $a$ is positive, the graph is an upwards facing symmetrical smooth curve with a minimum turning point (vertex); if $a$ is negative, the curve is downwards facing with a maximum turning point.

What is the formula for the $x$ coordinate of the vertex of a quadratic function?

The value of $x$ at the vertex is calculated using the formula $-\frac{b}{2a}$.

How are the $x$ and $y$ intercepts of a quadratic function determined?

The $x$ intercepts are found by solving for $x$ when $y = f(x) = 0$. The $y$ intercept is obtained by finding the value of the function when $x = 0$, or $y = f(0)$.

How many solutions are expected for linear versus quadratic equations?

There is always one solution for a linear equation, and there are two solutions for a quadratic equation.