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Linear graphs gradients and intercepts - Mathematics Year 8

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Introduction

Does a line through (1, 4) and (3, 10) have equation y=3 x+2 just because its gradient is 3? Find the signed rise and run, then recover the intercept instead of assuming it from a separate example.

You will calculate six gradients, identify four y-intercepts, plot a line from its rule and justify which distinct lines are parallel. A vertical line and a horizontal line need different treatment; axis units matter when interpreting rates.

Calculate six gradients from signed coordinate changes

For two points with different x-coordinates, gradient is signed vertical change divided by signed horizontal change: $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ . Keep the point order consistent in numerator and denominator. Reversing both differences leaves the quotient unchanged; reversing only one changes its sign incorrectly.

Original point-pair example. From (1, 4) to (3, 10), rise = 10−4 = 6 and run = 3−1 = 2, so gradient is 3. Reversing the direction gives (4−10)/(1−3)=−6/−2=3. This line has y=3 x+c; using (1, 4) gives 4=3+c, hence c=1. Therefore its equation is y=3 x+1. The separate source example y=3 x+2 has the same gradient but a different intercept and does not pass through these two points.

Six original gradients.

Point A	Point B	Signed rise	Signed run	Gradient
(1, 4)	(3, 10)	6	2	3
(0, 5)	(2, 1)	−4	2	−2
(−2, −1)	(2, 1)	2	4	1/2
(−3, 4)	(5, 4)	0	8	0
(1, −2)	(4, 7)	9	3	3
(−1, 6)	(3, −2)	−8	4	−2

Cover the final columns before calculating. A negative gradient means y decreases as x increases on these axes. A zero gradient means y is constant: the fourth pair lies on horizontal line y=4. The first and fifth rows have the same gradient despite different starting heights; the second and sixth likewise share gradient −2.

Vertical exception. Points (2, 1) and (2, 4) have run 0, so their gradient would require division by zero. It is undefined, not zero. Their line is x=2 and cannot be represented as y=mx+c for a single finite m. A horizontal line does fit that form with m=0.

If x and y measure different quantities, gradient carries y-units per x-unit. On purely numerical coordinate axes no invented physical unit is needed. Read scale labels before using the apparent steepness of a screenshot; equal pixel changes are not necessarily equal numerical changes. Two identical points do not determine one unique line. Choose distinct points and confirm that their x-values differ before using the quotient; a repeated point would give an indeterminate zero-over-zero calculation.

Identify four intercepts and compare parallel lines

The equation y=mx+c records two separate features: m is the gradient and c is the y-value when x=0. Its y-intercept point is (0, c). The first plotted point need not lie on the y-axis, so its y-coordinate is not automatically c. Substitute x=0 or use a known point and the gradient.

Four original intercepts.

Line rule	Gradient	y-intercept value	y-intercept point
y=3 x+2	3	2	(0, 2)
y=−2 x+5	−2	5	(0, 5)
y=(1/2)x −4	1/2	−4	(0, −4)
y=7	0	7	(0, 7)

For y=3 x+2, choose x=−1, 0, 1, 2 to obtain y=−1, 2, 5, 8. Plot the ordered pairs with consistent axis scales and draw the stated straight line. If the supplied grid does not fit y=8 at your chosen scale, redraw or relabel a suitable axis rather than clipping a point or changing the equation. A run of one x-unit gives a rise of three y-units; a run of two gives six.

Parallel lines require a common coordinate system. The first point-pair line y=3 x+1 and the fifth pair's line y=3 x −5 have equal gradient 3 and distinct intercepts, so they are parallel. The separate line y=3 x+2 is parallel to both. Lines with the same gradient and the same intercept coincide; they are not two distinct parallel lines. The second and sixth point-pair rows give y=−2 x+5 and y=−2 x+4, another distinct parallel pair.

Use Corbettmaths’ original gradient-between-two-points explanation to track coordinate differences. The video’s own point labels are separate from our six pairs; apply the method and then determine c from one point rather than assuming the intercept. Original: corbettmaths, Gradient between two points - Corbettmaths. The written explanations and native tasks work without sound or the external player.

Check a plotted line by substituting at least two listed points into its stated equation and by comparing signed rise/run. These checks help detect swapped coordinates, an intercept read from the wrong point or a scale error. A diagram alone at unknown zoom does not calibrate real-world physical distances.

Check line rules and diagnose coordinate errors

Two complete source examples, kept distinct. The point pair (1, 4), (3, 10) gives m=3 and c=1. Check 4=3×1+1 and 10=3×3+1. The separate rule y=3 x+2 gives m=3 and c=2. At x=1 its y-value is 5, not 4. Shared gradient does not imply that two lines are the same line.

A second way to find gradient. For a known rule y=3 x+2, compare values at x=0 and x=1: y changes from 2 to 5, giving rise 3 over run 1. Compare x=−1 and x=1 instead: y changes from −1 to 5 over run 2, again gradient 3. Algebraically, the constant cancels from differences: $\frac{(3 x_{2} + 2) - (3 x_{1} + 2)}{x_{2} - x_{1}} = 3 (x_{2} \neq x_{1})$ .

Error clinic. A learner uses (3−1)/(10−4)=1/3, reversing rise and run. Another uses (10−4)/(1−3)=−3, reversing only the x-order. A third calls 4 the intercept because the first supplied point is(1, 4). Correct the first changed operation or definition; the resulting line y=3 x+1 must satisfy both actual original points.

Horizontal and vertical cases supply useful checks. The points (−3, 4), (5, 4) give rise 0/run 8=0 and equation y=4. Points with the same x-coordinate give a zero denominator and a vertical line, not a zero gradient. Do not force that case into a finite y=mx+c rule.

Unfamiliar transfer. Points (−2, 7) and (2, −1) have rise −8 and run 4, giving m=−2. Use 7=(−2)(−2)+c to find c=3, hence y=−2 x+3. Verify both points: at −2 it gives 7 and at 2 it gives −1. It is parallel to y=−2 x+5 but has a different intercept. Comparing line slopes assumes the same axis quantities and unit scales.

Retain six signed gradient calculations, four intercepts, the complete y=3 x+2 plot and both parallel-line explanations. State which line each point belongs to and show a check that could expose the first error. An actual later unfamiliar pair after a genuine break is separate evidence; these ideal numerical plots do not certify physical measurement accuracy or long-term mastery. If only two observations are supplied, their connecting line is a model, not proof that every future observation follows it. Here the tasks explicitly ask about straight lines, so that assumption is stated rather than inferred from any two data values.

Interactive Tasks

Quiz: Test Your Knowledge

What is the gradient through the source points 1, 4 and 3, 10? (3) (!1/3) (!−3) (!6)

What is the equation through that source pair? (y=3 x+1) (!y=3 x+2) (!y=x+3) (!y=−3 x+7)

What is the y-intercept value of the separate line y=3 x+2? (2) (!3) (!1) (!5)

What is the gradient through 0, 5 and 2, 1? (−2) (!2) (!−1/2) (!−4)

What is the gradient through negative 3, 4 and 5, 4? (0) (!Undefined) (!4) (!8)

What is the gradient of the vertical line x=2? (Undefined) (!0) (!2) (!1/2)

What identifies two distinct parallel lines in the same coordinate system? (Equal gradients and different intercepts) (!Equal intercepts and different gradients) (!The same first plotted y-value) (!Both have positive intercepts)

What y-value does y=3 x+2 give at x=1? (5) (!4) (!3) (!2)

What is the y-intercept value of y=one half x minus four? (−4) (!1/2) (!4) (!0)

What line passes through negative 2, 7 and 2, negative 1? (y=−2 x+3) (!y=2 x+3) (!y=−2 x+5) (!y=−3 x+1)

Memory Game

Gradient	Signed vertical change divided by signed horizontal change
Intercept	Coordinate value where a graph crosses a specified axis
Parallel	Distinct coplanar lines here with the same gradient
Horizontal	Line with constant y and zero gradient
Vertical	Line with constant x and undefined finite gradient
Coordinate	Number in an ordered pair describing position

Drag and Drop

3	Gradient for points 1, 4 and 3, 10
1	Intercept of the line through that original pair
2	Intercept of the separate rule y=3 x+2
−2	Gradient for points 0, 5 and 2, 1
0	Gradient of the horizontal point pair

Crossword Puzzle

GRADIENT	Vertical change divided by horizontal change
INTERCEPT	Value where a line crosses a named axis
PARALLEL	Describes distinct lines with equal slopes here
HORIZONTAL	Line with constant y
VERTICAL	Line with constant x
COORDINATE	Value in an ordered position pair

LearningApps

Optional extra practice. Choose an appropriate activity with your teacher. If the external collection is unavailable, use the native activities and independent tasks in this course.

Cloze Text

Open Tasks

Easy

Calculate six slopes: Use every actual point pair and show signed rise and run.
Identify four intercepts: Complete all four intercept rows and state each point on the y-axis.
Separate two source lines: Explain why the point-pair example gives 3 x+1 ratherthan 3 x+2.
Distinguish special lines: Compare horizontal y 4 withverticalx 2 without dividing by zero.

Standard

Plot a line accurately: Generate four values for y 3 x+2 and draw a suitably scaled labelled plot.
Check with new point pairs: Find the same slope through different pairs on that stated line.
Explain parallel pairs: Derive the first/fifth and second/sixth point-pair intercepts and compare gradients.
Repair first errors: Correct reversed rise/run, inconsistent coordinate order and a false first-point intercept.

Advanced

Transfer to an unfamiliar pair: Find the rule through −2, 7 and 2, −1 with two substitutions.
Explain an identity for slope: Show why the constant cancels from the difference quotient for 3 x+2.
Evaluate a visual claim: Explain why screenshot steepness or two coincident equations do not establish two distinct parallel lines.
Retrieve later: Keep an actual later gradient/intercept/parallel task with signed differences, first correction and check.

Learning Check

Six exact gradients: Give the slope of each actual point pair in order.
Four intercepts and a plot: Identify the four intercept values and plot the first rule.
Source examples and parallelism: Keep the original pair separate from the named rule.
Horizontal, vertical and error analysis: Explain the two zero-change cases and three first errors.
Unfamiliar complete solution: Derive the rule through −2, 7 and 2, −1.
Evidence and later checking: Retain complete scope with meaningful units and checks.

Worked checks and assessment guidance

3, −2, 1/2, 0, 3, −2. Use signed rises 6/−4/2/0/9/−8 and runs 2/2/4/8/3/4. Reverse both coordinate differences together if choosing the opposite point order.
For 3 x+2, −2 x+5, (1/2)x −4 and 7, intercepts are 2, 5, −4, 7. First-rule points −1, −1; 0, 2; 1, 5; 2, 8 lie on its straight line with gradient 3.
Points 1, 4 and 3, 10 give y 3 x+1. The separate y 3 x+2 is parallel, not identical. Pairfivegives 3 x −5; pairtwo −2 x+5 andpairsix −2 x+4 alsoformdistinctparallel lines.
Horizontal 4 hasgradient 0; verticalx 2 hasrun 0 andundefinedgradient. Invertingrise/runoronlyonecoordinateorderchangesm; firstpointy 4 atx 1 isnotc. Originalpointsubstitutionexposesthewrongline.
m=(−1−7)/(2−(−2))=−8/4=−2; 7=4+c givesc 3, so y −2 x+3. Both original points check; this is parallel to −2 x+5 withdifferentintercept.
Keep six gradients, four intercepts, a labelled plot and parallel reasoning. Gradient units come from the named axes; blank grid and translation image are not measured physical maps. Actual later work and measurement skill need separate evidence.

Learning evidence

Retain six gradients, four intercepts, a labelled plot, two parallel-pair explanations and an unfamiliar full line equation with substitution checks. Distinguish the two separate source examples and actual later evidence.

OER and Wikipedia

Department for Education: England mathematics programme of study, Key Stage 3, algebra, graphs, fluency, reasoning and problem solving. Year 8 allocation is editorial within KS3. Numerical examples, tables, questions and worked feedback are original. Commons media retain their actual creators and licences below; original-publisher videos supplement the complete written teaching. Reviewed 12 November 2026.

2D Cartesian Coordinates.svg, H Padleckas, Public domain. Unchanged original; caption and learning activities are original.

Translation illustration.svg, Oleg Alexandrov, Public domain. Unchanged original; caption and learning activities are original.