Linear Graphs

Describing Straight Lines:

Drawing Linear equations

Drawing (xy table)
Drawing using $y = m x + c$: Practice Questions; Textbook Exercise

Vertical Lines:

x = k

where $k$ is some constant, e.g., $x = 3$ . This is the equation of a vertical straight line. The more positive $k$ is the more the vertical line is to the right. The more negative, the more the vertical line is to the left.

$x = a$ graphs: Practice Questions; Textbook Exercise

Horizontal Lines:

y = k

where $k$ is some constant, e.g., $y = -2$ . This is the equation of a horizontal straight line. The more positive $k$ is the higher the horizontal line is. The more negative, the lower the horizontal line is.

$y = a$ graphs: Practice Questions; Textbook Exercise

Lines at other Angles:

y = m x + c

where $m$ and $c$ are constants, i.e., $y = 2 x + 4$

$m$ is the gradient of the line.

$c$ is the y-axis intercept [ $\frac{-c}{m}$ is the x-axis intercept]

Gradient of a Line:

Gradient, m = \frac{change in y}{change in x} = \frac{Δ y}{Δ x} = \frac{y_{B} - y_{A}}{x_{B} - x_{A}}

Gradient of a line
Gradient between points: Practice Questions; Textbook Exercise

The Equation of a Line:

These are some of the ways the equation of a line maybe written:

y = m x + c

x - k = 0

y - k = 0

a x + b y + c = 0

a x + b y = c

As seen above, $y = m x + c$ is one of the most useful presentation.

Notice that both $x$ and $y$ have an index one, i.e., there are no other indices in any terms.

$y = m x + c$: Practice Questions; Textbook Exercise

Length of a line

Graph showing a line segment from point A to point B

l^{2} = {(x_{2} - x_{1})}^{2} + {(y_{2} - y_{1})}^{2}

Length of a line: Practice Questions; Textbook Exercise

Mid Point of a Line:

M : (x_{M}, y_{M}) = (\frac{x_{A} + x_{B}}{2}, \frac{y_{A} + y_{B}}{2})

Midpoint of a line: Practice Questions; Textbook Exercise

In real life
Across one method: Practice Questions; Textbook Exercise
Finding the equation of a line: Practice Questions; Textbook Exercise
Equation through 2 points: Practice Questions; Textbook Exercise

Comparing Lines: Parallel and Perpendicular Lines

For two or more lines are three possibilities for any pair. Lines $l_{1}$ and $l_{2}$ have equations $y = m_{1} x + c_{1}$ and $y = m_{2} x + c_{2}$ can either be parallel to each other (at an angle of 0° to each other), perpendicular to each other (at 90° to each other), or some other angle.

For parallel lines the gradients are equal, i.e., $m_{1} = m_{2}$ .

Parallel lines: Practice Questions; Textbook Exercise

For perpendicular lines the product of the gradients of the lines equal minus one, i.e., $m_{1} m_{2} = -1$

Perpendicular lines: Practice Questions; Textbook Exercise

Equation of a Perpendicular Bisector

To find the perpendicular bisector of a line segment between two given points:: Find the gradient of the line segment between the two points;; Find the gradient of the perpendicular using $m_{1} m_{2} = -1$ ;; Find the mid point of the two given points. The perpendicular bisector goes through this mid point;; To find the y-intercept of the perpendicular bisector substitute the mid point coordinates in the equation for the bisector.

Linear Graphs(Straight Lines)