Why Start Here?#
The linear function is the simplest function in mathematics. It is a straight line. If you deeply understand how each parameter controls a straight line, you will find it much easier to understand the same parameters in parabolas, hyperbolas, and exponentials.
1. The Two Forms#
Gradient-Intercept Form#
$$ y = mx + c $$- $m$ = The gradient (slope). It tells you how steep the line is and which direction it faces.
- $c$ = The y-intercept. It tells you where the line crosses the y-axis.
Turning-Point Style (for consistency with other functions)#
$$ y = a(x - p) + q $$This is the same thing, just written differently:
- $a$ = The gradient (same as $m$).
- $p$ = Horizontal shift.
- $q$ = Vertical shift.
For a straight line, $y = a(x - p) + q$ simplifies back to $y = ax - ap + q = ax + (q - ap)$, which is just $y = mx + c$ where $c = q - ap$.
2. The Parameters: What Each One Does#
The Gradient ($m$ or $a$)#
The gradient controls two things: steepness and direction.
| Value of $m$ | Effect |
|---|---|
| $m > 0$ | Line slopes upward from left to right (increasing function) |
| $m < 0$ | Line slopes downward from left to right (decreasing function) |
| $m = 0$ | Line is perfectly horizontal ($y = c$) |
| $m$ undefined | Line is perfectly vertical ($x = k$) — not a function |
| $ | m |
| $ | m |
The “Rise over Run” Logic: $m = \frac{\Delta y}{\Delta x} = \frac{y_2 - y_1}{x_2 - x_1}$
If $m = 3$, it means: for every 1 unit you move right, you move 3 units up.
The y-Intercept ($c$)#
| Value of $c$ | Effect |
|---|---|
| $c > 0$ | Line crosses the y-axis above the origin |
| $c < 0$ | Line crosses the y-axis below the origin |
| $c = 0$ | Line passes through the origin |
Think of $c$ as a vertical shift. Changing $c$ slides the entire line up or down without changing its slope.
3. Key Properties#
| Property | Value |
|---|---|
| Domain | $x \in \mathbb{R}$ (all real numbers) |
| Range | $y \in \mathbb{R}$ (all real numbers) |
| x-intercept | Set $y = 0$: $x = -\frac{c}{m}$ |
| y-intercept | Set $x = 0$: $y = c$ |
| Asymptotes | None |
| Function type | One-to-One (every output has exactly one input) |
4. Finding the Equation#
There are several scenarios you will encounter in exams:
Given the gradient and y-intercept#
Substitute directly into $y = mx + c$.
Example: $m = 2$, $c = -3$
$$ y = 2x - 3 $$Given the gradient and one point#
Use the point-gradient form: $y - y_1 = m(x - x_1)$
Example: $m = -1$, point $(4; 3)$
$$ y - 3 = -1(x - 4) $$$$ y = -x + 4 + 3 $$$$ y = -x + 7 $$Given two points#
- Calculate $m = \frac{y_2 - y_1}{x_2 - x_1}$
- Use point-gradient form with either point.
Example: Points $(1; 5)$ and $(3; 11)$
$$ m = \frac{11 - 5}{3 - 1} = \frac{6}{2} = 3 $$$$ y - 5 = 3(x - 1) $$$$ y = 3x + 2 $$5. The Inverse of a Linear Function#
Because a linear function is One-to-One, its inverse is always a function (no domain restriction needed).
The “Swap” Method#
- Start with $y = 2x + 6$
- Swap $x$ and $y$: $x = 2y + 6$
- Solve for $y$: $$ 2y = x - 6 $$ $$ y = \frac{x - 6}{2} $$ $$ y = \frac{1}{2}x - 3 $$
- Write: $f^{-1}(x) = \frac{1}{2}x - 3$
What Happens to the Parameters?#
| Original $f(x) = mx + c$ | Inverse $f^{-1}(x) = \frac{1}{m}x - \frac{c}{m}$ |
|---|---|
| Gradient = $m$ | Gradient = $\frac{1}{m}$ (reciprocal) |
| y-intercept = $c$ | y-intercept = $-\frac{c}{m}$ |
| Domain: $x \in \mathbb{R}$ | Domain: $x \in \mathbb{R}$ |
| Range: $y \in \mathbb{R}$ | Range: $y \in \mathbb{R}$ |
Key Insight: The gradient of the inverse is the reciprocal of the original gradient (not the negative reciprocal—that’s for perpendicular lines, which is a different concept).
Graphical Check#
The graphs of $f$ and $f^{-1}$ are reflections of each other across the line $y = x$. If the point $(2; 10)$ is on $f$, then $(10; 2)$ must be on $f^{-1}$.
6. Special Cases#
Horizontal Line: $y = c$#
- Gradient = 0.
- This is a Many-to-One function (every $x$ gives the same $y$).
- Its inverse ($x = c$) is a vertical line, which is not a function.
Vertical Line: $x = k$#
- This is not a function at all (fails the Vertical Line Test).
- It has no inverse.
The line $y = x$#
- This is its own inverse! Swapping $x$ and $y$ gives you $x = y$, which is the same line.
- Every function and its inverse intersect on this line.
Worked Example: Full Exam-Style Question#
Given $f(x) = -3x + 6$:
(a) Determine the x-intercept and y-intercept of $f$.
x-intercept (set $y = 0$):
$$ 0 = -3x + 6 $$$$ 3x = 6 $$$$ x = 2 $$x-intercept: $(2; 0)$
y-intercept (set $x = 0$):
$$ y = -3(0) + 6 = 6 $$y-intercept: $(0; 6)$
(b) Determine $f^{-1}(x)$.
Swap $x$ and $y$:
$$ x = -3y + 6 $$$$ 3y = 6 - x $$$$ y = -\frac{1}{3}x + 2 $$$$ f^{-1}(x) = -\frac{1}{3}x + 2 $$(c) Verify that $(6; -12)$ is on $f$, and find the corresponding point on $f^{-1}$.
$$ f(6) = -3(6) + 6 = -18 + 6 = -12 \checkmark $$Corresponding point on $f^{-1}$: Swap the coordinates → $(-12; 6)$.
Check: $f^{-1}(-12) = -\frac{1}{3}(-12) + 2 = 4 + 2 = 6 \checkmark$
🚨 Common Mistakes#
- Confusing gradient and y-intercept: In $y = 5 - 2x$, the gradient is $-2$ (not $5$). Always rearrange to $y = -2x + 5$ first.
- Inverse gradient error: The inverse of $y = 3x$ is $y = \frac{1}{3}x$, NOT $y = -\frac{1}{3}x$. Don’t confuse inverse with perpendicular.
- Parallel vs Equal: Two lines with the same gradient are parallel, not the same line (unless $c$ is also equal).
💡 Pro Tip: Reading the Graph#
In an exam, if they give you the graph and ask for the equation:
- Read $c$ directly from where the line crosses the y-axis.
- Calculate $m$ by picking two clear grid points and using $\frac{\text{rise}}{\text{run}}$.
- Write the equation. Done.
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