Straight-line graphs and what they mean
A straight-line graph turns a rule into a picture. Gradient and intercept are not just numbers on a page: in a real context they are a rate and a starting value.
Part of your national curriculum
- Algebra: Work with coordinates in all four quadrants and recognise, sketch and produce graphs of linear functions
Lesson overview
What you'll learn in this lesson
Work with coordinates in all four quadrants and recognise, sketch and produce graphs of linear functions
Key learning points
- • y = mx + c
- • Finding gradient from two points
- • Parallel and negative gradients
- • Reading real meaning
This lesson at a glance
- 30 minutes
- 25 parts to scroll through
- 4 quick checks
- Marked quiz at the end
- Gentle pace: short sittings with pauses
Words to know
Scroll down — the lesson carries on below
Watch & discover
Part 1 of 25
Visual introductionPicture this
Straight-line graphs and what they mean
A straight-line graph turns a rule into a picture. Gradient and intercept are not just numbers on a page: in a real context they are a rate and a starting value.
In a nutshell
Work with coordinates in all four quadrants and recognise, sketch and produce graphs of linear functions
What you already know
Part 2 of 25
Before we start
What you already know
You should already be able to plot coordinates in all four quadrants and substitute values into an expression.
Key words
Part 3 of 25
Key words
Words you'll need today
- Gradient
- How steep a line is: change in y divided by change in x.
- Intercept
- Where the line crosses the y-axis.
- Linear
- Making a straight line when plotted.
- Parallel
- Lines with the same gradient that never meet.
- Rate of change
- What the gradient means in a real context.
Reset break
Part 4 of 25
Pause
That's sitting 1 of 6 done
Stretch, get a drink, look out of the window. There is no timer and nothing is counting down — your place is saved, so you can come back in five minutes or tomorrow.
Learning cycle
Part 5 of 25
Learning cycle 1 of 2
Reading the equation
A short piece of teaching, then a check to make sure it has landed.
Explore the idea
Part 6 of 25
Learn
y = mx + c
In y = mx + c the gradient m tells you how much y changes for each increase of one in x, and c is where the line crosses the y-axis. For y = 3x − 4 the line rises 3 for every 1 across, and crosses the y-axis at (0, −4).
Explore the idea
Part 7 of 25
Learn
Finding gradient from two points
Gradient is the change in y divided by the change in x. Through (1, 5) and (4, 14) the gradient is (14 − 5) ÷ (4 − 1) = 3. Substituting one point back gives 5 = 3(1) + c, so c = 2 and the equation is y = 3x + 2.
Reset break
Part 8 of 25
Pause
That's sitting 2 of 6 done
Stretch, get a drink, look out of the window. There is no timer and nothing is counting down — your place is saved, so you can come back in five minutes or tomorrow.
Quick check
Part 9 of 25
Quick check
Part 10 of 25
Learning cycle
Part 11 of 25
Learning cycle 2 of 2
Comparing lines and meaning
A short piece of teaching, then a check to make sure it has landed.
Reset break
Part 12 of 25
Pause
That's sitting 3 of 6 done
Stretch, get a drink, look out of the window. There is no timer and nothing is counting down — your place is saved, so you can come back in five minutes or tomorrow.
Explore the idea
Part 13 of 25
Learn
Parallel and negative gradients
Parallel lines have equal gradients and different intercepts. A negative gradient means y falls as x rises, so y = −2x + 7 slopes downwards. A horizontal line has gradient zero and equation y = a constant; a vertical line has equation x = a constant and no defined gradient.
Explore the idea
Part 14 of 25
Learn
Reading real meaning
For a taxi fare F = 2.5d + 3, the gradient 2.5 is the cost per mile and the intercept 3 is the fixed charge before travelling. Answering 'what does the gradient represent' with a number alone gains nothing: it must be described in the units of the context.
Quick check
Part 15 of 25
Reset break
Part 16 of 25
Pause
That's sitting 4 of 6 done
Stretch, get a drink, look out of the window. There is no timer and nothing is counting down — your place is saved, so you can come back in five minutes or tomorrow.
Quick check
Part 17 of 25
Explore the idea
Part 18 of 25
Worked example
Worked answer: find the equation of the line through (1, 5) and (4, 14) (3 marks)
The gradient is (14 − 5)/(4 − 1) = 9/3 = 3 (1). Substituting (1, 5) into y = 3x + c gives 5 = 3 + c, so c = 2 (1). The equation is y = 3x + 2, which checks against the second point: 3(4) + 2 = 14 (1).
Common mix-ups
Part 19 of 25
Common mix-ups
Things people often get wrong
- People often read the gradient as just the change in y. It is the change in y for every one step in x.
- People often think a negative gradient means the line is below the axis. It means the line falls from left to right.
Reset break
Part 20 of 25
Pause
That's sitting 5 of 6 done
Stretch, get a drink, look out of the window. There is no timer and nothing is counting down — your place is saved, so you can come back in five minutes or tomorrow.
Challenge round
Part 21 of 25
Game · Sort it
Which of these are true?
Drag each card into the right column. Tap a card first if dragging is fiddly.
True
Not true
Challenge round
Part 22 of 25
Game · Recall cards
Which point lies in quadrant III?
Card 1 of 4
Exit quiz
Part 23 of 25
Exit quiz
Show what you've learned
6 questions, marked together at the end. Nothing is timed.
1. In y = 3x − 2, the gradient is...
2. In y = 3x − 2, the line crosses the y-axis at...
3. A line through (1, 2) and (3, 8) has gradient...
4. Which line is parallel to y = 2x + 1?
5. A negative gradient means the line...
6. On a distance-time graph, the gradient represents...
Mastery quiz
Part 24 of 25
Marked quiz
End of lesson quiz: Straight-line graphs and what they mean
4 questions, marked with the reasoning shown. No timer.
1. Which point lies in quadrant III?
2. For y=2x+1, find y when x=3.
3. What is the gradient of y=−3x+4?
4. What is the gradient of y = 3x − 4?
Lesson round-up
Part 25 of 25
Lesson round-up
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Work with coordinates in all four quadrants and recognise, sketch and produce graphs of linear functions