Sec 1 Linear Graphs: Worked Examples & Practice

Secondary 1 Mathematics · G3 focus · Study guide

Plot ordered pairs, build straight-line graphs, interpret gradient and intercepts, and explore how changing a linear rule changes its graph.

Concept → worked examples → practice15 min read · practice linked below

Coordinates connect numbers to a position

Two perpendicular number lines form the coordinate axes. The horizontal axis is the $x$ -axis; the vertical axis is the $y$ -axis. Their intersection is the origin $(0,0)$ . An ordered pair $(x,y)$ tells you to move horizontally first, then vertically. The order matters: $(2,-3)$ and $(-3,2)$ are different points.

A point on the $y$ -axis has $x=0$ ; a point on the $x$ -axis has $y=0$ . In the upper-right quadrant both coordinates are positive. In the upper-left they are negative, positive; in the lower-left both are negative; in the lower-right they are positive, negative.

Worked example 1: plot and describe a point

Plot $P=(-2,3)$ and $Q=(2,-3)$ . State which quadrant contains each point.

For $P$ , start at the origin, move 2 units left, then 3 units up. It lies in the upper-left quadrant.
For $Q$ , move 2 units right, then 3 units down. It lies in the lower-right quadrant.
Read each point back in horizontal-then-vertical order: $P=(-2,3)$ and $Q=(2,-3)$ .

Check: neither point lies on an axis because neither coordinate is zero. Their opposite coordinates put them opposite each other through the origin.

Each marked horizontal or vertical unit represents the same distance. Dashed paths show the coordinate movements.

A linear rule makes a straight line

A relationship can assign a $y$ -value to each $x$ -value. In $y=2x+1$ , multiply the input $x$ by 2, then add 1 to get the output $y$ . The points $(x,y)$ that obey the rule lie on a straight line.

From rule to graph

Choose useful $x$ -values, calculate $y$ , and record each pair in a table. Draw labelled axes with an even scale on each axis. Plot at least three points to catch calculation errors, then use a ruler to draw the line over the required domain. A continuous line represents all permitted intermediate $x$ -values. If the situation only allows whole-number inputs, plot separate points instead.

Worked example 2: make and check a table

Draw $y=2x+1$ for $-2\leq x\leq2$ , then find $y$ when $x=0.5$ .

Substitute each $x$ into the rule. For example, when $x=-2$ , $y=2(-2)+1=-3$ .
The coordinate pairs are $(-2,-3),\ (-1,-1),\ (0,1),\ (1,3),\ (2,5).$ Plot them and join with one straight segment from $(-2,-3)$ to $(2,5)$ .
Read upward from $x=0.5$ to the line, then across to the $y$ -axis. This gives $y=2$ .

Check algebraically: $2(0.5)+1=\boxed{2}$ . When $x$ increases by 1, $y$ increases by 2. If a plotted point breaks that pattern, check its calculation or position.

Read a value from the plotted segment

This is the segment required in worked example 2. Its filled endpoints include both −2 and 2. The dashed path reads the intermediate point

(0.5,2)

Explore gradient and intercept

In $y=mx+c$ , $m$ is the gradient and $c$ is the $y$ -intercept: the value of $y$ where $x=0$ . A positive gradient rises from left to right; a negative gradient falls. Zero gradient gives a horizontal line.

The initial line is $y=2x+1$ . It crosses the $y$ -axis at 1 and rises 2 units for each 1-unit move to the right. Change the gradient and intercept, then move the marked input.

Read the numbers on each axis: one horizontal grid interval is 1 unit, while one vertical grid interval is 5 units. Gradient uses coordinate changes, not the apparent angle on the screen.

At x = 1, y = 2 × 1 + 1 = 3. A move of 1 to the right changes y by +2. The y-intercept is (0, 1).

Measure gradient using coordinate changes

Choose two distinct points on the line. Compute the change in $y$ and the change in $x$ in the same direction: $m=\frac{\text{vertical change}}{\text{horizontal change}}=\frac{y_2-y_1}{x_2-x_1}.$ A horizontal line has zero vertical change, hence gradient 0. A vertical line has zero horizontal change: division by zero is undefined, so its gradient is undefined. Its equation has the form $x=k$ , not $y=mx+c$ .

Worked example 3: a falling line

A straight line passes through $A=(-2,5)$ and $B=(2,-1)$ . Find its gradient.

Move from $A$ to $B$ . The horizontal change is $2-(-2)=4$ .
The vertical change is $-1-5=-6$ , a downward change.
Divide in the correct order: $m=\frac{-6}{4}=\boxed{-\frac32}.$

The line falls 3 units for every 2 units moved to the right. Check in the reverse direction: $(5-(-1))/(-2-2)=6/(-4)=-3/2$ , the same gradient. Both subtraction orders must reverse together.

Worked example 4: interpret a graph in context

A bicycle rental costs a fixed $3 plus $2 per hour. Part-hours are charged proportionally. Write the cost $C$ for $t$ hours, find the cost for 2.5 hours, and explain the graph's gradient and intercept.

The time charge is $2t$ dollars. Add the fixed charge: $C=2t+3,\qquad t\geq0.$
At $t=2.5$ , $C=2(2.5)+3=\boxed{8}.$ The cost is $8.
The gradient is 2 dollars per hour: each additional hour adds $2. The vertical intercept is 3 dollars, the fixed charge.

Check: 2 hours cost $7 and 3 hours cost $9, so 2.5 hours cost $8. Draw only the part with $t\geq0$ , because negative rental time has no meaning. Label the horizontal axis “time (hours)” and the vertical axis “cost ($)”.

Common mistakes and further practice

Reversing coordinates: $(x,y)$ means horizontal first, vertical second.
Uneven scales: equal numerical changes on one axis need equal spacing along that axis.
Using the appearance of steepness: count coordinate changes, especially when the axes have different scales.
Reading the wrong intercept: the $y$ -intercept is found with $x=0$ ; the $x$ -intercept is found with $y=0$ .
Joining every data set: use a continuous line only when intermediate inputs are permitted by the rule and situation.

For related Sec 1 algebra skills, use the linear equations practice bank. For practical applications, use the Sec 1 rate, speed and time bank. For further graph questions, the Sec 2 linear and quadratic graphs bank includes more advanced work; begin with its straight-line questions.