Graphs of linear functions: intercepts and gradient – Linear Functions, Equations and Inequalities (Senior 1 Mathematics) | Ganzaa

Linear Functions, Equations and Inequalities

Lessons

Senior 1 · Mathematics

Linear Functions, Equations and Inequalities

5 lessons

Lesson 2 of 5

Graphs of linear functions: intercepts and gradient

25 min

Suppose a moto rider in Kigali charges a fixed 300 Frw to start a trip, plus 100 Frw for every kilometre. The fare depends on the distance, and each extra kilometre adds exactly the same amount. Put that into a table and something useful appears.

Distance $x$ (km)	0	1	2	3	4	5
Fare $y$ (Frw)	300	400	500	600	700	800

The fare rises by 100 Frw at every step, so the points $(x, y)$ lie on a straight line. The rule behind the table is $y = 100 x + 300$ .

Definition

Function: a rule that gives exactly one output for each input. We write $f (x)$ , read " $f$ of $x$ ", for the output when the input is $x$ .

Example: if $f (x) = 100 x + 300$ , then $f (2) = 100 (2) + 300 = 500$ .

Definition

Linear function: a function of the form $f(x) = mx + c \quad\text{or}\quad y = mx + c,$ where $m$ and $c$ are fixed numbers. Its graph is a straight line. $m$ is the gradient and $c$ is the $y$ -intercept.

Example: in $y = 100 x + 300$ , $m = 100$ and $c = 300$ .

1. Drawing the graph from a table of values

Choose at least three values of $x$ . Three points let you catch a mistake: if they are not in a straight line, recheck your arithmetic.
Work out $y$ for each, and set the results in a table.
Draw and label the axes with a suitable scale, plot the points, and join them with a ruler. Extend the line past the points and label it with its equation.

Worked example

Draw the graph of $y = 2 x - 1$ for $-2 \le x \le 3$

Substitute each value of $x$ . For example, when $x = - 2$ : $y = 2 (- 2) - 1 = - 5$ .

$x$	$- 2$	$- 1$	$0$	$1$	$2$	$3$
$y = 2 x - 1$	$- 5$	$- 3$	$- 1$	$1$	$3$	$5$

Plot the six points and join them. The line crosses the $y$ -axis at $(0, - 1)$ , and $c = - 1$ as the equation predicts.

2. Intercepts

Definition

$y$ -intercept: the point where a graph crosses the $y$ -axis. Every point on the $y$ -axis has $x = 0$ , so find it by putting $x = 0$ .

$x$ -intercept: the point where a graph crosses the $x$ -axis. Every point on the $x$ -axis has $y = 0$ , so find it by putting $y = 0$ .

Example: for $y = 2 x - 1$ , the $y$ -intercept is $(0, - 1)$ and the $x$ -intercept is $(\tfrac{1}{2}, 0)$ .

Worked example

Find the intercepts of $2 x + 3 y = 6$ and use them to draw the line

$y$ -intercept: put $x = 0$ . Then $3 y = 6$ , so $y = 2$ . The point is $(0, 2)$ .
$x$ -intercept: put $y = 0$ . Then $2 x = 6$ , so $x = 3$ . The point is $(3, 0)$ .
Check with a third point: $x = - 1.5$ gives $3 y = 6 + 3 = 9$ , so $y = 3$ . The point $(- 1.5, 3)$ lies on the same straight line.

3. Gradient (slope)

Definition

Gradient (or slope) of a straight line: how steep it is, measured as the change in $y$ for each change in $x$ . For any two points $(x_{1}, y_{1})$ and $(x_{2}, y_{2})$ on the line, $m = \frac{\text{rise}}{\text{run}} = \frac{y_2 - y_1}{x_2 - x_1}.$

Example: on $y = 2 x - 1$ , from $(0, - 1)$ to $(2, 3)$ the rise is 4 and the run is 2, so $m = \frac{4}{2} = 2$ .

Why the gradient is the same everywhere on the line. Draw a gradient triangle anywhere under a straight line. Each such triangle has the same angles, so they are enlargements of one another, and $\frac{\text{rise}}{\text{run}}$ is the same for all of them. That single number is $m$ in $y = m x + c$ : each step of 1 to the right changes $y$ by $m$ .

Gradient	The line…	Example
$m > 0$	rises from left to right	$y = x + 1$
$m \lt 0$	falls from left to right	$y = - 2 x + 3$
$m = 0$	is horizontal	$y = - 2$

Worked example

Find the gradient of the line through $A (- 1, 5)$ and $B (3, - 3)$ , then its equation

m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{-3 - 5}{3 - (-1)} = \frac{-8}{4} = -2.

The line falls, as a negative gradient should. So $y = - 2 x + c$ . Point $A$ is on the line, so $5 = - 2 (- 1) + c$ , which gives $c = 3$ . The equation is $y = - 2 x + 3$ : the orange line in the diagram above.

Worked example

Use the moto graph. What do $m$ and $c$ mean, and how far can you ride for 1,000 Frw?

$c = 300$ : the fare when $x = 0$ , the fixed charge to start a trip.
$m = 100$ : the fare rises 100 Frw for each kilometre. The gradient is a rate, 100 Frw per km.
Find 1,000 on the fare axis, go across to the line and down: $x = 7$ . Check: $100 (7) + 300 = 1,000$ . So 1,000 Frw pays for 7 km.

Warning

Keep the same order top and bottom. In $\frac{y_2 - y_1}{x_2 - x_1}$ , if you start with point $B$ on top you must start with $B$ underneath too. Mixing the order flips the sign of the gradient. And the formula is rise over run: $\frac{\text{run}}{\text{rise}}$ is upside down.

Warning

Read $m$ as the number multiplying $x$ . In $y = 5 - 2 x$ the gradient is $- 2$ , not 5. Rewrite it as $y = - 2 x + 5$ first if that helps.

Tip

In $y = m x + c$ , think "m for move" (how much $y$ moves for each step across) and "c for crossing" (where the line crosses the $y$ -axis).

Math Antics (mathantics): "Basic Linear Functions". Watch for how a table of values becomes a straight line, and how the slope and the y-intercept show up in y = mx + b (b is another letter for c).

✦

Practice

Draw your own axes on squared paper where a graph is needed.

Complete a table of values for $y = 3 x - 2$ for $x = - 1, 0, 1, 2, 3$ .
State the gradient and the $y$ -intercept of (a) $y = 4 x + 7$ ; (b) $y = 5 - 2 x$ ; (c) $y = \frac{1}{2}x$ .
Does the point $(3, 8)$ lie on $y = 2 x + 1$ ?
Find the $x$ - and $y$ -intercepts of $3 x + 4 y = 12$ .
Find the gradient of the line through $(1, 2)$ and $(4, 11)$ .
Find the gradient of the line through $(- 2, 6)$ and $(4, 3)$ . Does the line rise or fall?
A plumber in Musanze charges a 2,000 Frw call-out fee plus 1,500 Frw per hour. Write the charge $y$ for $x$ hours. State the gradient and the $y$ -intercept, and what each means.
A line has gradient $- 3$ and passes through $(0, 4)$ . Write its equation and find its $x$ -intercept.

Show answersHide answers

$y = - 5, - 2, 1, 4, 7$
(a) $m = 4,\ c = 7$ ; (b) $m = -2,\ c = 5$ ; (c) $m = \frac{1}{2},\ c = 0$ (the line passes through the origin).
No: $2(3) + 1 = 7 \ne 8$ .
$(4, 0)$ and $(0, 3)$ .
$m = \frac{11 - 2}{4 - 1} = \frac{9}{3} = 3$
$m = \frac{3 - 6}{4 - (-2)} = \frac{-3}{6} = -\frac{1}{2}$ . It falls.
$y = 1,500 x + 2,000$ . Gradient 1,500: the charge rises 1,500 Frw per hour. $y$ -intercept 2,000: the call-out fee, paid even for 0 hours.
$y = - 3 x + 4$ . Put $y = 0$ : $3 x = 4$ , so $x = \frac{4}{3}$ . The $x$ -intercept is $(\frac{4}{3}, 0)$ .

Sources and credits

4.2: Graph Linear Equations in Two Variables, from Elementary Algebra by Lynn Marecek, MaryAnne Anthony-Smith and Andrea Honeycutt Mathis, OpenStax, provided by LibreTexts, licensed under CC BY 4.0. Adapted by Ganzaa.
4.3: Graph with Intercepts, from Elementary Algebra by Lynn Marecek, MaryAnne Anthony-Smith and Andrea Honeycutt Mathis, OpenStax, provided by LibreTexts, licensed under CC BY 4.0. Adapted by Ganzaa.
4.4: Understanding the Slope of a Line, from Elementary Algebra by Lynn Marecek, MaryAnne Anthony-Smith and Andrea Honeycutt Mathis, OpenStax, provided by LibreTexts, licensed under CC BY 4.0. Adapted by Ganzaa.
Grade 8, Unit 3: Linear Relationships, IM 6–8 Math by Illustrative Mathematics, originally developed by Open Up Resources (© 2017–2019 Open Up Resources; adaptations © 2019 Illustrative Mathematics), provided by Kendall Hunt, licensed under CC BY 4.0. Adapted by Ganzaa.
Photo: A Moto (motorcycle taxi) in Kigali.jpg, Jenny Paul, Wikimedia Commons, licensed under CC BY-SA 4.0. Used unchanged.
Video: Basic Linear Functions, mathantics, embedded from YouTube.

The moto-fare model, the worked examples, diagrams, practice items and quiz questions were written by Ganzaa. The fare figures are a model for practice, not an official tariff.

This lesson includes CC BY-SA material and is shared under CC BY-SA 4.0.

11. Drawing the graph from a table of values

22. Intercepts

33. Gradient (slope)

1. Drawing the graph from a table of values

2. Intercepts

3. Gradient (slope)