Maths Revision

Gradient-intercept form of a linear equation

$y = mx + c$, where $m$ represents the gradient (slope) and $c$ represents the $y$-intercept.

Formula for the gradient ($m$) of a straight line

$$m = \frac{\text{rise}}{\text{run}} = \frac{y<em>2 - y</em>1}{x<em>2 - x</em>1}$$

Algebraic method to determine the $y$-intercept

Substitute $x = 0$ into the rule and solve for $y$.

Algebraic method to determine the $x$-intercept

Substitute $y = 0$ into the rule and solve for $x$.

Rule and $y$-intercept of a straight line passing through the origin

The $y$-intercept is $0$, and the equation takes the direct proportional form $y = mx$.

Visual trend of a line with a positive versus negative gradient

A line with a positive gradient rises from left to right, whereas a line with a negative gradient falls from left to right.

Condition for two straight lines to be parallel

Both lines must share the exact same gradient ($m<em>1 = m</em>2$).

Physical quantity represented by the gradient of a distance-time graph

Speed ($\text{speed} = \frac{\text{distance}}{\text{time}}$).

Meaning of a horizontal (flat) segment on a distance-time graph

The object has stopped moving (stationary), as elapsed time increases while distance remains constant.

Formula for average speed

$$\text{Average Speed} = \frac{\text{Total Distance}}{\text{Total Time}}$$

Method for calculating a unit rate to determine the best buy

Divide the total price by the quantity ($\frac{\text{cost}}{\text{quantity}}$) to obtain cost per unit; the lower value gives the better buy.

Real-world meaning of the intersection point of two linear cost graphs

The specific value of the independent variable where both options cost the exact same amount.

Algebraic check to verify an intersection point of two equations

Substitute the coordinates of the ordered pair $(x, y)$ into each separate rule; the point is verified if both equations hold true.

Real-world meaning of $m$ and $c$ in a cost model $C = mx + c$

$m$ represents the variable rate (cost per unit), while $c$ represents the fixed fee or initial start-up cost.