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9.2 Linear Graphs

GCSE Maths Foundation · Edexcel · Chapter 9: Graphs

A free GCSE Maths Foundation lesson for Edexcel (9–1) from OBA School: lines parallel to the axes, y = x and y = −x, tables of values and plotting straight lines. Watch the video, read the explanation and worked examples, then try the revision questions. You can work through the full lesson, with interactive activities, practice quizzes and printable worksheets, in our free online course.

What you will learn

  • Recognise and draw lines such as x = 3, y = −2, y = x and y = −x
  • Complete a table of values for an equation such as y = 2x + 1
  • Plot a straight-line graph accurately from a table of values
  • Check whether a point lies on a line

Key words

  • Equation of a line: A rule connecting x and y that is true for every point on the line, e.g. y = 2x + 1.
  • Linear: A linear graph is a straight line. Its equation has no x² or other powers.
  • Table of values: A table of x values and the matching y values worked out from the equation.
  • Horizontal / vertical: Horizontal goes across, like the horizon. Vertical goes straight up and down.

Lines parallel to the axes

Every point on a vertical line has the same x-coordinate. The points (3, −1), (3, 1) and (3, 2) all have x = 3, so they lie on the line x = 3. Every point on a horizontal line has the same y-coordinate, for example y = −2.

Key fact: x = a is a vertical line through a on the x-axis. y = b is a horizontal line through b on the y-axis. The x-axis is the line y = 0 and the y-axis is the line x = 0.

Worked example: naming a line

A line passes through (4, 1), (4, 6) and (4, −3). What is its equation?
The x-coordinate is always 4, so the line is vertical.

Answer: x = 4

The lines y = x and y = −x

On y = x the two coordinates are equal: (1, 1), (2, 2), (−3, −3). On y = −x the y-coordinate is the negative of the x-coordinate: (1, −1), (−3, 3). Both lines pass through the origin.

Tables of values

To draw a sloping line such as y = 2x + 1, choose some x values, substitute each one into the equation and work out y. Take care with negative numbers.

Worked example: y = 2x + 1

x = −2: y = 2 × (−2) + 1 = −4 + 1 = −3. x = 0: y = 2 × 0 + 1 = 1. x = 3: y = 2 × 3 + 1 = 7.

Answer: see the completed table (drawn in the printable lesson notes)

  • y = 2x + 1; −2 −3; −1 −1; 0 1; 1 3; 2 5; 3 7

Check: In a linear table the y values go up (or down) by the same amount each time. Here they go up by 2. A value that breaks the pattern is a mistake.

Plotting a straight-line graph

Plot each pair (x, y) from the table as a point. The points should lie in a straight line. Join them with a ruler and draw the line across the whole grid, not just between the first and last points. Label the line with its equation.

Worked example: equations like x + y = 6

Find two easy points. When x = 0, y = 6. When y = 0, x = 6.
Plot (0, 6) and (6, 0), check with a third point such as (2, 4) since 2 + 4 = 6, and join them.

Answer: the teal line on the grid

Is a point on the line?

Substitute the x-coordinate into the equation. If you get the y-coordinate, the point is on the line.

Worked example: checking points

Is (5, 11) on y = 2x + 1? 2 × 5 + 1 = 11. Yes.
Is (−4, −9) on y = 2x + 1? 2 × (−4) + 1 = −7, not −9. No.

Answer: (5, 11) is on the line; (−4, −9) is not

Common mistakes

  • Drawing x = 3 as a horizontal line. x = 3 is vertical: every point on it has x-coordinate 3.
  • Sign slips with negative x, e.g. 2 × (−2) + 1 = 5. It is −4 + 1 = −3.
  • Joining the points “dot to dot” freehand. Use a ruler and draw one straight line.
  • Stopping the line at the end points. Extend it across the grid you are given.
  • Plotting (y, x) instead of (x, y) from the table.

Revision questions

Try these, then check your answers below each one.

Question 1. What is the equation of the vertical line through the point (5, −2)? (Choose from: x = 5, y = 5, y = -2, x = -2.)

Answer: x = 5

Working: All points on the line have x = 5, so the equation is x = 5.

Question 2. For the line y = 4x - 3, work out the value of y when x = -2.

Answer: -11

Working: y = 4 × (-2) - 3 = -8 - 3 = -11.

Question 3. Kai says, “The point (3, 7) lies on the line y = 2x + 1.” Is Kai correct?

Answer: Yes

Working: 2 × 3 + 1 = 7, so (3, 7) is on the line. Kai is correct.

Related lessons

Continue in the full course

This lesson is section 9.2 (EGCMF0902) of our free GCSE Maths Foundation course. In the full course you get every video for this section, an animated interactive lesson, a self-marking practice quiz, a printable worksheet with a mark scheme, and your progress saved. Go to the full GCSE Maths Foundation course.

 

 

 

 

 

 

 

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