Liri had learned to plot single points. But now she wanted to describe a whole straight flight path – not just one point.
Erklärus: “That is what linear functions are for. A single formula describes a whole straight line.”
A formula for a straight line
📈 y = mx + b – every number has a meaning
The formula has exactly two adjusting screws: m and b. Each of them controls one particular part of the line.
m – the gradient
Erklärus: “What happens when x gets 1 bigger? y changes by m – that is the gradient.”
Liri: “So m=2 means: for every step to the right the line goes up by 2. And m=−1 means it falls by 1?”
Erklärus: “Exactly. And m=0? Then y never changes – the line is horizontal.”
m > 0
rising ↗
m < 0
falling ↘
m = 0
horizontal →
|m| large
steep ↑
b – the y-intercept
Erklärus: “Where does the line cross the y-axis? Substitute x=0: y = m×0 + b = b. The y-intercept is simply b.”
Liri: “So b=3 means: the line starts at y=3 on the y-axis, no matter how steep it is?”
Erklärus: “Exactly. And b=0 means: the line goes through the origin (0, 0).”
substitute x=0 → y=b → crossing point (0, b)
Function y = 3x + 2 – how big is y when x=4?
📊 Drawing a straight line – two points are enough
To draw a straight line you only need two points. Then you draw the line through them – done.
Erklärus: “Draw y = 2x + 1. Which two points do you choose?”
Liri: “I take x=0 and x=3. For x=0: y=2×0+1=1 → point (0, 1). For x=3: y=2×3+1=7 → point (3, 7). Plot both, draw the line – done!”
Table of values for y = 2x + 1
x
−2
−1
0
1
y
−3
−1
1
3
Gradient m=2: y grows steadily by 2 for every x increase of 1 → that gives a straight line ✓
🧮 Liri’s function plotter – interactive
Change m and b and watch how the line changes – the quickest way to understand:
📈 Linear function – y = mx + b
m =
b =
y = 2x + 1
📍 The root – where does the line cross the x-axis?
The root (x-intercept) is the point where the line crosses the x-axis – there y = 0. You find it by setting y=0 and solving for x.
Liri: “The root of y=2x+1? I set y=0: 0=2x+1. That is an equation! Minus 1: −1=2x. Divided by 2: x=−0.5.”
Erklärus: “The line crosses the x-axis at x=−0.5. And do you see what you have just done? You solved an equation – just like in chapter 43. Algebra and geometry are closely related.”
📍 Important points of a line
y-intercept: substitute x=0 → y=b → point (0, b)
Root: set y=0 → 0=mx+b → x = −b ÷ m
Reading off the gradient: m = Δy ÷ Δx
Root of y = 4x − 8: x =
🌌 Linear functions in the universe
The height of a rocket during a steady climb can often be described by a linear function: height = speed × time.
Engineers use such functions to predict where an object is at a particular moment in time.
❌ Common mistake
The gradient (m) and the y-intercept (b) are often swapped!
Tip: b is the starting value at x=0 (where the line meets the y-axis). m tells you how steeply it goes up or down from there.
📈 Exercise 1
Function: y = 4x + 3. How big is y when x=2?
📊 Exercise 2
For the function y = 5x - 2: what y-value does the function have when x=0?
🚀 Exercise 3 – Universe
A rocket climbs according to y = 50x (height = 50 × time). How high is it after x=6 seconds?
A whole line, a single formula
Liri described her whole straight flight path with a single function equation – every point of the journey was now contained in a single formula.
Liri: “I no longer need every point separately – one formula describes my whole course.”
“Linear function: y = mx + b. m = gradient, b = y-intercept (starting value at x=0). The graph is always a straight line.”
What you learned today
y = mx + b
m = gradient · b = y-intercept
🌟
Chapter 47 mastered!
Liri can now describe a whole straight flight path with a single formula.
Erklärus’s notebook – page 47:
“Linear function: y=mx+b. m=gradient (how steep), b=y-intercept (starting value at x=0). The graph is always a straight line.”
“Linear function: y=mx+b. m=gradient (how steep), b=y-intercept (starting value at x=0). The graph is always a straight line.”