Linear Functions & Graphs

Linear functions connect algebra to visual patterns. The equation tells you how $y$ changes as $x$ changes, and the graph shows that relationship as a straight line.

In CSEC, graph questions may ask for gradient, intercepts, equations of lines, parallel or perpendicular lines, or graphical solutions to simultaneous equations. Do not only plot points; explain what the gradient and intercept mean. That helps with comprehension and reasoning marks.

What Is a Linear Function?

A linear function is a function where the graph is a straight line.

General form: $f(x) = mx + c$

Or: $y = mx + c$

Where:

• $m$ = slope (gradient), how steep the line is
• $c$ = y-intercept, where the line crosses the y-axis
• $x$ = input (domain variable)
• $y$ or $f(x)$ = output (range variable)

Understanding Slope (Gradient)

Slope measures how much $y$ changes when $x$ increases by 1.

$m = \text{slope} = \frac{\text{change in } y}{\text{change in } x} = \frac{\Delta y}{\Delta x} = \frac{y_2 - y_1}{x_2 - x_1}$

Interpretation:

• $m > 0$: Line goes UP from left to right (positive slope)
• $m < 0$: Line goes DOWN from left to right (negative slope)
• $m = 0$: Horizontal line (flat, no change)
• $m$ undefined: Vertical line

Examples of Different Slopes

Forms of Linear Equations

Form 1: Slope-Intercept Form (Most Useful)

$y = mx + c$

• Easy to identify: slope is $m$, y-intercept is $c$
• Easy to graph: plot $(0, c)$, then use slope to find more points

Form 2: Point-Slope Form

$y - y_1 = m(x - x_1)$

Use this when you know:

• The slope $m$
• One point $(x_1, y_1)$ on the line

Form 3: Two-Point Form

When you know two points $(x_1, y_1)$ and $(x_2, y_2)$:

Step 1: Find slope $m = \frac{y_2 - y_1}{x_2 - x_1}$

Step 2: Use point-slope form with either point

Form 4: Standard Form

$Ax + By + C = 0$

Or: $Ax + By = C$

Where $A$, $B$, $C$ are integers with no common factors.

Convert from slope-intercept to standard: $y = 2x - 3 \Rightarrow 2x - y - 3 = 0$

Finding Intercepts

Y-Intercept

The y-intercept is where the line crosses the y-axis (when $x = 0$).

To find: Set $x = 0$ and solve for $y$.

X-Intercept

The x-intercept is where the line crosses the x-axis (when $y = 0$).

To find: Set $y = 0$ and solve for $x$.

Properties of Linear Functions

Parallel Lines

Two lines are parallel if they have the same slope and different y-intercepts.

$y = m x + c_1 \text{ and } y = m x + c_2 \text{ (where } c_1 \neq c_2)$

Parallel lines never intersect.

Perpendicular Lines

Two lines are perpendicular if their slopes are negative reciprocals of each other.

If line 1 has slope $m_1$ and line 2 has slope $m_2$: $m_1 \times m_2 = -1$

Or: $m_2 = -\frac{1}{m_1}$

Perpendicular lines meet at a 90° angle.

Examples:

• Slope 2 and slope $-\frac{1}{2}$ are perpendicular (because $2 \times (-\frac{1}{2}) = -1$)
• Slope 3 and slope $-\frac{1}{3}$ are perpendicular
• Slope $\frac{3}{4}$ and slope $-\frac{4}{3}$ are perpendicular

Length and Midpoint of Line Segments

Distance (Length) Formula

For two points $(x_1, y_1)$ and $(x_2, y_2)$, the distance between them is:

$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

This comes from the Pythagorean theorem.

Midpoint Formula

The midpoint of a line segment between $(x_1, y_1)$ and $(x_2, y_2)$ is:

$\text{Midpoint} = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)$

Just average the x-coordinates and average the y-coordinates.

Graphing Linear Functions

Sketching from Slope-Intercept Form

Given $y = mx + c$:

Step 1: Plot the y-intercept $(0, c)$

Step 2: Use slope $m = \frac{\text{rise}}{\text{run}}$ to find more points:

• If $m = 2 = \frac{2}{1}$: Go right 1, up 2
• If $m = -3 = \frac{-3}{1}$: Go right 1, down 3
• If $m = \frac{2}{3}$: Go right 3, up 2

Step 3: Plot at least 3 points and draw the line through them

Solving Systems Graphically

When you have two linear equations, the solution is the point where the lines intersect.