Graph a line from a table of values, read the gradient and y-intercept straight out of $y = mx + c$, and use a line to model something real.
Today’s hook: In $y = mx + c$ the $m$ is the steepness and the $c$ is where the line crosses the y-axis. Read those two numbers and you can sketch the line without plotting a single point.
Before diving into the practice questions, take a moment to think about what you already know about this topic.
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1
What You'll Master
objectives
Know
A linear equation $y = mx + c$ produces a straight-line graph
Understand
$m$ is the gradient, it controls the steepness and direction of the line
$c$ is the y-intercept, it shows where the line crosses the y-axis
Can Do
Complete a table of values for a linear equation
Plot and connect points to draw a straight-line graph
Read the gradient and y-intercept from the equation $y = mx + c$
2
Words You Need
vocabulary
Linear RelationshipA relationship between two quantities that forms a straight line when graphed. The rule can always be written as $y = mx + c$.
Gradient (Slope)How steep a line is. A positive gradient means the line rises left to right; a negative gradient means it falls. Calculated as rise ÷ run.
y-interceptThe point where the line crosses the y-axis. It happens when $x = 0$. In $y = mx + c$, the y-intercept is $c$.
Table of ValuesA table where you substitute chosen $x$ values into a rule to find the matching $y$ values. Gives you coordinates to plot.
$y = mx + c$The standard form of a linear equation. $m$ is the gradient, $c$ is the y-intercept, $x$ is the input, and $y$ is the output.
Rise over RunThe formula for gradient: $\text{gradient} = \dfrac{\text{rise}}{\text{run}} = \dfrac{\text{change in }y}{\text{change in }x}$. Rise = vertical change, run = horizontal change.
3
Table of Values Method
+5 XP to read
To graph a linear equation, substitute a set of $x$ values to find the matching $y$ values. Each pair $(x, y)$ is a point you can plot on the Cartesian plane.
Plot each $(x, y)$ pair on the Cartesian plane, then connect them with a straight line.
$y = 2x + 1$
2 points minimum
You only need 2 points to draw a straight line, but use 3 or more to catch arithmetic mistakes.
Use $x = 0$ first
Starting with $x = 0$ gives you the y-intercept straight away, the easiest point to plot.
Points in a line?
If your 3 points don't line up, one calculation is wrong, go back and check each substitution.
4
Gradient and y-intercept from $y = mx + c$
+5 XP to read
Once you know the form $y = mx + c$, you can read off the gradient and y-intercept instantly, no table needed.
In $y = mx + c$: $m$ is the gradient and $c$ is the y-intercept.
For $y = 3x - 2$:
Gradient $= 3$ (rises 3 units for every 1 unit right)
y-intercept $= -2$ (line crosses y-axis at $-2$)
Positive gradient → line slopes up (left to right). Negative gradient → line slopes down.
$$\text{Gradient} = \frac{\text{rise}}{\text{run}} = \frac{\text{change in }y}{\text{change in }x}$$
$y = \underbrace{3}_{m}x \underbrace{- 2}_{c}$
$m$ comes first
In $y = mx + c$, the number in front of $x$ is always $m$. In $y = 3x - 2$, it's 3, not $-2$.
$c$ is the stand-alone number
$c$ is the number not multiplied by $x$. Watch signs: in $y = 4x - 7$, $c = -7$, not 7.
Steeper = larger $|m|$
A gradient of 5 is steeper than a gradient of 2. The bigger the absolute value, the steeper the line.
5
Real-World Linear Models
+5 XP to read
Linear equations appear everywhere in real life. Any situation with a fixed starting amount and a constant rate of change can be modelled by $y = mx + c$.
Scenario: A plumber charges an $\$80$ call-out fee plus $\$60$ per hour.
$C = 60h + 80$
Gradient $= 60$ → cost per extra hour
y-intercept $= 80$ → fixed call-out fee
Hours ($h$)
0
1
2
3
4
Cost ($C$)
$\$80$
$\$140$
$\$200$
$\$260$
$\$320$
$C = 60h + 80$
Gradient = rate of change
In real-world models, the gradient is always the rate: dollars per hour, metres per second, etc.
y-intercept = starting value
The y-intercept is the value when $x = 0$, the fixed fee, starting balance, or initial amount.
Watch Me Solve It · Worked example
Watch Me Solve It · Graphing $y = -x + 3$
+15 XP per step
Q
PROBLEM
Graph $y = -x + 3$ for $x = 0, 1, 2, 3$. Identify the gradient, y-intercept, and x-intercept.
In $y = mx + c$: $m$ is the gradient (number in front of $x$) and $c$ is the y-intercept (the stand-alone number), not the other way around. In $y = 4x + 7$, the gradient is 4 and the y-intercept is 7.
Fix: ask "what's multiplying $x$?" That's $m$. Then ask "what's left by itself?" That's $c$.
Using only 2 points then making an error
A line only needs 2 points to define it, but if either calculation has an arithmetic error, you'll draw the wrong line. Using 3 or more points gives you a built-in check, if they're not collinear, you've made a mistake.
Fix: always calculate at least 3 points and verify they form a straight line before connecting them.
Calculating gradient as run ÷ rise
Gradient = rise ÷ run, not run ÷ rise. Rise is the vertical change ($y$), run is the horizontal change ($x$). Writing it upside-down gives the wrong gradient every time.
Fix: remember "rise over run", rise (up/down, $y$) is on top; run (left/right, $x$) is on the bottom. $m = \dfrac{\Delta y}{\Delta x}$
Quick Check · 5 questions
1
Which table matches $y = 3x + 2$?
+10 XP
2
A plumber charges $40 callout plus $30 per hour. Which rule gives the cost $C$ for $h$ hours?
+10 XP
3
For $y = 2x + 5$, what is the y-intercept?
+10 XP
4
A graph passes through $(0, 3)$ and $(2, 7)$. What is its equation?
+10 XP
5
From the graph of $y = 3x + 1$, what is $y$ when $x = 5$?
+10 XP
ApplyEasy2 MARKS
Q1. Complete the table of values for $y = 4x - 2$, then describe the pattern in the $y$ values.
Answer in your workbook.
ApplyMedium3 MARKS
Q2. A gym membership costs $\$20$ joining fee plus $\$12$ per week. Write the rule and calculate the cost for 15 weeks.
Answer in your workbook.
ApplyHard4 MARKS
Q3. A line passes through $(1, 5)$ and $(3, 11)$. Find the slope and y-intercept, and write the equation of the line.
Answer in your workbook.
Stretch Challenge · +25 XP, +10 coins
Extension Problems
A car rental costs $\$50$ per day plus $\$0.30$ per kilometre. Write a rule and find the cost for driving ...
R
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Interactive: Algebra Machine
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