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Sec 2 Linear Graphs | Gradient, y-Intercept and Straight-Line Equations

Secondary 2 Linear Graphs connects algebraic equations to visible relationships on the coordinate plane. A straight-line graph is not merely a set of plotted points; it is a geometric picture of how one variable changes with another.

Students searching for Sec 2 linear graphs, gradient, y-intercept, straight-line graph or Secondary 2 Mathematics often learn to plot tables without seeing the algebra underneath. The more durable system is equation ↔ table ↔ points ↔ line ↔ gradient/intercept ↔ interpretation.

This page is the canonical Linear Graphs owner under the Secondary Mathematics Topic Library. Its Sec 1 dependencies are Linear Equations and Number Patterns.

Quick answer: the straight-line form

y = mx + c

  • m: gradient or slope.
  • c: y-intercept, the value of y when x = 0.

Coordinates

A point (x,y) tells you horizontal position first, vertical position second.

Example: (3,−2) means move 3 units right and 2 units down from the origin.

Plotting from a table

Equation: y = 2x + 1.

For x = −1,0,1,2, the y-values are −1,1,3,5.

Plot the points and join them with a straight line.

Gradient as rate of change

Gradient = change in y ÷ change in x.

For points (1,3) and (4,9): gradient = (9−3)/(4−1) = 6/3 = 2.

A positive gradient rises left to right; a negative gradient falls.

Worked example: find gradient

Points A(−2,5) and B(2,−3).

m = (−3−5)/(2−(−2)) = −8/4 = −2.

The y-intercept

The y-intercept is where the line crosses the y-axis, so x=0.

For y = 3x − 4, the y-intercept is −4 and the line passes through (0,−4).

Reading gradient and intercept from an equation

In y = −1.5x + 6, gradient = −1.5 and y-intercept = 6.

Students should be able to predict the graph’s broad shape before plotting.

Finding the equation of a line

  1. Find the gradient.
  2. Write y = mx + c.
  3. Substitute one known point to find c.
  4. Write the final equation.
  5. Check with another point if available.

Worked example: equation from two points

Line passes through (1,4) and (3,10).

Gradient = (10−4)/(3−1) = 3.

So y = 3x + c.

Use (1,4): 4 = 3 + c, so c = 1.

Equation: y = 3x + 1.

Horizontal and vertical lines

Horizontal line: y = constant, gradient 0.

Vertical line: x = constant. Its gradient is undefined because change in x is zero.

This distinction becomes important when students later study perpendicular and parallel lines.

Parallel lines

Parallel non-vertical straight lines have the same gradient.

Example: y=2x+1 and y=2x−5 are parallel because both have gradient 2.

Intersection of two lines

The intersection point satisfies both equations at the same time.

Graphically, it is where the lines cross. Algebraically, it is the solution of the corresponding simultaneous equations.

This is the bridge to Sec 2 Simultaneous Equations.

Graph interpretation

A graph can represent cost, distance, temperature or another changing quantity.

Gradient then carries context: dollars per item, kilometres per hour, degrees per minute, and so on.

The intercept often represents a starting value, fixed charge or initial state.

Worked context example

Taxi fare C = 0.8d + 4, where d is distance in km.

Gradient 0.8 means $0.80 per km.

Intercept 4 means a $4 starting charge.

Common linear-graph errors

  • reverses x and y coordinates
  • uses run/rise instead of rise/run
  • forgets negative signs in gradient
  • reads c as x-intercept instead of y-intercept
  • plots points against uneven axis scales
  • joins points inaccurately
  • finds equation without checking a known point

The graph audit

  1. Are axes labelled?
  2. Is the scale consistent?
  3. Does the point order use (x,y)?
  4. Does the line direction match the sign of the gradient?
  5. Does the line cross the y-axis at c?
  6. Does a known point satisfy the equation?

Practice

Practice 1

Question: y=4x−3: gradient

Answer: 4

Practice 2

Question: y=4x−3: y-intercept

Answer: −3

Practice 3

Question: Gradient through (1,2),(5,10)

Answer: 2

Practice 4

Question: Equation gradient 2 through (0,5)

Answer: y=2x+5

Practice 5

Question: Line y=−3

Answer: horizontal

Practice 6

Question: Line x=4

Answer: vertical

Frequently asked questions

What does gradient mean?

Change in y per unit change in x.

What is the y-intercept?

The y-value where x=0.

How do I find a line equation from two points?

Find gradient, use y=mx+c, then substitute a point to find c.

How are graphs linked to simultaneous equations?

The solution of two linear equations is the intersection of their lines.

Where does this sit in Atlas?

This is the canonical Sec 2 Linear Graphs owner under Secondary Mathematics Topic Library.

The final Linear Graph rule

See the equation and graph as two representations of one relationship. Gradient describes change; intercept describes starting position. Once those meanings are stable, plotting and interpretation become much easier.

Explore the connected learning guides

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A word is familiar, but using it is difficult.

Move from recognising a word to retrieving it in a new context. Understand vocabulary plateaus.

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Make the order of events and the links between sentences clear. Explore composition writing.

Try it without the guide: Choose one short paragraph. Read the relevant explanation, close it, and revise the paragraph. Ask someone to tell you what happened and why.

The Mathematics seems familiar, but marks still disappear.

Find the first point where the working stops being reliable. Find Secondary 4 A-Math mark leakage.

Try it without the guide: For a Secondary 4 A-Math question you have attempted, locate the first uncertain line. Repair that step, then try a comparable question without the worked answer.

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Connect the evidence to a scientific idea and the resulting change. Follow the Primary Science learning route.

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Check the question, source, date and evidence before combining claims. Explore the World Knowledge research library.

Try it without the guide: Take one claim. Find the source best placed to support it, note its date, and state what remains uncertain. Return to your original question.

There is plenty of help, but independence is hard to see.

Check what the learner can understand and do after support is removed. Understand how education works.

Try it without the guide: Choose one small task the child has practised. Agree on a calm, brief attempt without prompts. Use what happens to choose one next step, then stop.

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