Transform equations into visual lines on the coordinate plane. Master the table method and the efficient gradient-intercept technique.
45 min 5 MCQ + 3 SAQ MAS-LIN-C-01
Think First
Before we begin: Without drawing, how many points do you think you need to plot to draw a straight line accurately? Explain why.
Come back to this after you have worked through the lesson.
1
The Big Idea
Every linear equation produces a straight line when graphed on the coordinate plane. Two points are enough to define a unique straight line, but plotting three gives a useful check. The table of values method works for any equation, while the gradient-intercept method is faster when the equation is already in $y = mx + c$ form.
Know
The two main methods for graphing linear functions. What x-intercept and y-intercept mean.
Understand
Why only two points are needed to define a straight line. How gradient and y-intercept determine the position of a line.
Can Do
Graph a linear function using a table of values. Graph using the gradient-intercept method. Identify x and y intercepts.
2
Lesson Objectives
Complete a table of values for any linear equation.
Plot points accurately and draw a straight line through them.
Use the y-intercept and gradient to sketch a line quickly.
Find the x-intercept and y-intercept of a linear function.
3
Key Vocabulary
Table of values
A table showing x-values and their corresponding y-values for an equation.
Gradient-intercept method
Graphing using the y-intercept $(0, c)$ and the gradient $m$.
x-intercept
The point where a graph crosses the x-axis ($y = 0$).
y-intercept
The point where a graph crosses the y-axis ($x = 0$).
4
Spot the Trap
Wrong: Plotting only one point for a linear graph. A single point does not determine a unique line.
Right: Always plot at least two points (three is better for checking). Use a ruler to draw the line.
Wrong: Using $\dfrac{\text{run}}{\text{rise}}$ instead of $\dfrac{\text{rise}}{\text{run}}$ when moving from the y-intercept.
Right: From the y-intercept, move run units across and rise units up (or down if negative).
5
Graphing Using a Table of Values
+5 XP
The most reliable method for graphing any linear equation is to calculate points and plot them. Choose at least three x-values (including negative, zero, and positive), substitute each into the equation to find y, plot the points, and draw a straight line through them with a ruler.
Choose x-values. Calculate y. Plot and draw.
reliable for any equation
Always include x = 0
This gives the y-intercept directly. It is the easiest point to calculate.
Choose easy x-values
Pick numbers that make mental arithmetic simple. Avoid fractions if possible.
Three points for safety
Two points define a line, but three let you spot calculation errors. If they do not line up, recheck.
6
The Gradient-Intercept Method
+5 XP
When the equation is in $y = mx + c$ form, you can graph it quickly without a full table. Plot the y-intercept at $(0, c)$, then use the gradient $m = \dfrac{\text{rise}}{\text{run}}$ to find a second point. Move run units right and rise units up (or down if negative), then draw the line.
Plot $(0, c)$. Move run right, rise up. Draw line.
fast when in y = mx + c
Start at the y-intercept
$(0, c)$ is always on the y-axis. It is your anchor point.
Move run then rise
Go right by the run amount, then up (or down) by the rise amount. Do not reverse.
Extend both directions
A line continues forever. Draw arrows at both ends or extend beyond your plotted points.
7
Finding Intercepts
+5 XP
The x-intercept is where the line crosses the x-axis ($y = 0$). The y-intercept is where the line crosses the y-axis ($x = 0$). In $y = mx + c$ form, the y-intercept is simply $(0, c)$. To find the x-intercept, substitute $y = 0$ and solve for $x$.
x-intercept: $y = 0$. y-intercept: $x = 0$.
where the line crosses
y-intercept is easiest
In $y = mx + c$, the y-intercept is $(0, c)$. No calculation needed.
x-intercept: set y = 0
Substitute $y = 0$ into the equation and solve for $x$.
Both intercepts on axes
The x-intercept has y-coordinate 0. The y-intercept has x-coordinate 0.
8
Real-World Linear Relationships
+5 XP
Linear relationships appear everywhere: taxi fares (flat fee plus rate per km), phone bills (base cost plus per-minute charges), water usage (fixed service fee plus per-litre rate). In each case, the y-intercept represents the fixed cost and the gradient represents the variable rate.
y-intercept = fixed cost. gradient = rate.
C = 0.3d + 50
Fixed vs variable
The y-intercept is what you pay even if you use nothing. The gradient is what you pay per unit.
Break-even point
The x-intercept shows where total cost becomes zero (if it makes sense in context).
Label your axes
In real-world graphs, always label what x and y represent, including units.
To graph $y = -\frac{3}{4}x + 2$, from the y-intercept you move:
4 right, 3 down
Practice
5 MCQs and 3 short-answer questions. Target: 80% accuracy.
1.The y-intercept of $y = 3x - 5$ is:
Correct! In $y = mx + c$ form, $c$ is the y-intercept. Here $c = -5$, so the y-intercept is $(0, -5)$.
The y-intercept is the constant term in $y = mx + c$. It is not the coefficient of $x$.
2.Which point lies on the line $y = 2x + 3$?
Correct! Substitute $x = 2$: $y = 2(2) + 3 = 7$. So $(2, 7)$ lies on the line.
Substitute the x-coordinate into the equation and check if you get the y-coordinate. For example, for $(2, 7)$: $y = 2(2) + 3 = 7$.
3.The x-intercept of $y = 4x - 8$ is:
Correct! Set $y = 0$: $0 = 4x - 8$ → $4x = 8$ → $x = 2$. The x-intercept is $(2, 0)$.
The x-intercept occurs where $y = 0$. Substitute $y = 0$ into the equation and solve for $x$.
4.To graph $y = -\dfrac{3}{4}x + 2$ using gradient-intercept, from the y-intercept you should:
Correct! The gradient is $-\dfrac{3}{4}$, so rise = $-3$ (down 3) and run = $4$ (right 4). Move 4 right, then 3 down.
For $m = \dfrac{\text{rise}}{\text{run}} = -\dfrac{3}{4}$, the run is 4 (denominator) and the rise is $-3$ (negative means down). Move run units right first, then rise units up or down.
5.A line with gradient $-2$ passing through $(0, 5)$ has equation:
Correct! The gradient is $m = -2$ and the y-intercept is $c = 5$. So $y = -2x + 5$.
In $y = mx + c$, $m$ is the gradient and $c$ is the y-intercept. The point $(0, 5)$ tells you $c = 5$.
6.Complete a table of values for $y = -2x + 5$ using $x = -2, -1, 0, 1, 2$. Plot the points and describe the pattern you observe.
Your answer:
Correct table of values
Pattern described correctly
Connection to gradient made
Model Answer
| $x$ | $-2$ | $-1$ | $0$ | $1$ | $2$ |
| $y$ | $9$ | $7$ | $5$ | $3$ | $1$ |
The points form a straight line sloping downward from left to right. As $x$ increases by 1, $y$ decreases by 2 (consistent with gradient $-2$).
7.A line passes through $(0, -3)$ and has gradient $\dfrac{2}{5}$. (a) Write the equation of the line. (b) Find the x-intercept. (c) Explain how you would use the gradient-intercept method to sketch this line.
(c) Plot $(0, -3)$. From there, move 5 units right and 2 units up to $(5, -1)$. Draw line through both points.
8.A car rental company charges a $50 flat fee plus $0.30 per kilometre driven. Let $C$ be the total cost and $d$ be the distance in kilometres. (a) Write a linear equation relating $C$ and $d$. (b) Complete a table of values for $d = 0, 50, 100, 200$. (c) Graph the relationship, labelling the axes and the y-intercept. (d) Explain what the gradient and y-intercept represent in this context.
(c) Horizontal axis: distance (km). Vertical axis: cost ($). Plot points and draw straight line. y-intercept at $(0, 50)$.
(d) The gradient ($0.30$) represents the cost per kilometre (30 cents per km). The y-intercept ($50$) represents the fixed base fee regardless of distance.
Review
Consolidate and reflect before moving on.
Stretch Challenge
A line has x-intercept $(4, 0)$ and y-intercept $(0, -6)$. Find its equation in the form $y = mx + c$. Then find another point on this line that has integer coordinates. (Hint: find the gradient first using the two intercepts as points.)
Key Idea
Two methods for graphing linear functions: table of values (works for any equation) and gradient-intercept (fastest when in $y = mx + c$). Two points define a line; three confirm accuracy.
Common Trap
Plotting only one point. Reversing rise and run in the gradient-intercept method. Forgetting that a line extends infinitely in both directions.
Connection
The gradient-intercept method connects directly to the equation form $y = mx + c$. The next lessons will use these skills to find equations from graphs and to explore parallel and perpendicular lines.
Interactive: Line Grapher — enter any linear equation and see the graph, intercepts, and gradient visualised instantly.
Table Master
Gradient Sketcher
Intercept Finder
Daily Challenge
Sketch $y = -\dfrac{2}{3}x + 4$ using the gradient-intercept method. Label the y-intercept, x-intercept, and one other point. Time yourself — can you do it in under 60 seconds?
Printable Worksheets
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