Graphs | Notion

Linear Equation: an equation where a term is either a constant or a constant times a variable raised to the first power
Unit rate: rate at which $y$ changes with respect to $x$, i.e slope/gradient
The slope of a line is its gradient; larger the gradient, steeper the slope

$$ m = \dfrac{\Delta y}{\Delta x} = \dfrac{\text{'rise'}}{\text{'run'}} $$

Straight lines have no gradient
Parallel lines have the same gradient
A straight line graph can be displayed in the slope-intercept form, where $b$ is the $y$-intercept

$$ y = mx + b $$

A straight line graph can also take the standard form for linear equations. When written in standard form, $a, b$ must be integers where fractions are allowed in slope-intercept form

$$ ax + by = c $$

Standard form excels at finding the $x$ and $y$ intercepts
A straight line graph can be proven to be a functional relationship if it only has one intercept, if you can draw a vertical line across the relationship and it intersects more than one point on the line, it is not functional. A function $f(x)$ can only have one output for each input of $x$
Reading from a graph is not an exact science so values garnered may not be accurate

Functions

A function takes one input, produces one constant output: $f(x)$
The difference between an equation and a function is that equations deal with equivalences ($x + 3 = 10$) and functions focus on the relationships between variables.
- A function can be an equation
- A function is a special case of a relationship, not all relationships are functions
- A function does not need to be numerical (see: programming)
Functions can be linear and non-linear
- Linear: For any given change in $x$, the change in $y$ will always be constant. No values have exponents (except to the power of 1)
- Non-linear: not a straight line, one or more variables have exponents

An inequality is the relationship between two expressions that are not equal to each other, using $\leq$ and $\geq$
A system of inequalities is a set of two or more inequalities