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Cambridge IGCSE0580

Drawing linear graphs

Mathematics 0580 Chapter Notes

What this chapter covers

Drawing linear graphs
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Drawing linear graphs notes

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In this chapter

  1. 1Table of values
  2. 2Gradient and y-intercept
  3. 3Lines with gradient 0
  4. 4Sketching a line
  5. 5Equations not in the form y = mx + c

1. Table of values

A straight-line graph can be drawn from its equation. Choose values of x, work out each y, and plot the points (x, y).

Two points fix a straight line. A third point is a check. Join the points with one ruled straight line, across entire width of grid for the range of x the question gives.

Examiner insight

“Most errors were due to an incorrect sign but a few candidates gave entries apart from 7’s and 3’s.” “However, a significant number did not know it had to be a single straight line.”

Worked example 13 marks

Complete the table of values for y = 3x - 2. Then draw the graph for -2 ≤ x ≤ 2.

x-202
y
  1. 1

    x = -2: y = 3 × (-2) - 2 = -8

  2. 2

    x = 0: y = -2. x = 2: y = 3 × 2 - 2 = 4

  3. 3

    table: -8, -2, 4 [1]

  4. 4

    plot (-2, -8), (0, -2) and (2, 4) [1]

  5. 5

    join with one ruled straight line from x = -2 to x = 2 [1]

2. Gradient and y-intercept

When the equation is in the form y = mx + c, no table is needed. m is the gradient and c is the y-intercept: the line crosses the y-axis at (0, c).

Start at (0, c). Find a second point, such as the end of the range. Rule the line through both.

y = mx + c

gradient = (change in y)/(change in x)

Examiner insight

“Many candidates clearly did not know the meanings of ‘m’ and ‘c’ in the equation y = mx + c.” “Other incorrect lines included y = 0.5, y = 3.5 and lines that joined (–4, 0) to (0, 4) for example, making reference to the numbers appearing within the question.”

Worked example 12 marks

On the grid, draw the graph of y = 0.5x + 3 for -4 ≤ x ≤ 4.

  1. 1

    c = 3, so the line passes through (0, 3)

  2. 2

    x = -4: y = 0.5 × (-4) + 3 = 1. x = 4: y = 0.5 × 4 + 3 = 5

  3. 3

    ruled line through (-4, 1), (0, 3) and (4, 5), across the whole width of the grid [2]

3. Lines with gradient 0

A line with gradient 0 has m = 0. Its equation is y = 0x + c, which is y = c. The line is horizontal.

Every point on the line has the same y-coordinate. The number in the equation is the y-intercept, not the gradient.

m = 0 ⇒ y = c

Examiner insight

“Common incorrect answers were y = x + 7 and y = 7x + 0”

Worked example 11 mark

Write down the equation of the line that passes through the point (0, 7) and has gradient 0.

  1. 1

    gradient 0, so m = 0. The line crosses the y-axis at 7, so c = 7: y = 7 (or y = 0x + 7) [1]

4. Sketching a line

A sketch has no scale and needs no table. It shows the straight line with the right direction of slope and the right place where it crosses the y-axis.

A positive m slopes up from left to right. A negative m slopes down. A positive c is above the origin and a negative c is below it.

Worked example 11 mark

On the diagram, sketch the graph of y = 3x - 2.

  1. 1

    m = 3 is positive, so the line slopes up. c = -2, so it crosses the y-axis below the origin: a straight line with positive gradient and negative y-intercept [1]

5. Equations not in the form y = mx + c

Some equations, such as y = 7 - 4x and 3x + 2y = 5, are not written as y = mx + c. Rearrange so that y is on its own. Then read off m and c.

For y = 7 - 4x, write y = -4x + 7. The gradient is -4 and the y-intercept is 7.

An equation such as x = 8 has no y in it. Every point on the line has x-coordinate 8, so the line is vertical. Each line is drawn as a ruled straight line.

3x + 2y = 5 ⇒ y = -3/2x + 5/2

Examiner insight

“The line x = 15 produced the least number of errors followed by y = 50, which was sometimes drawn as x = 50.”

Worked example 12 marks

On the grid, draw the line y - 2x + 3 = 0.

  1. 1

    rearrange: y = 2x - 3, so c = -3 and the line passes through (0, -3)

  2. 2

    x = 3: y = 2 × 3 - 3 = 3, so (3, 3) is on the line

  3. 3

    ruled correct line at least from (0, -3) to (3, 3) [2]

Recap

  • Straight line from an equation: table of values, plot, join with one ruled line.
  • Draw across the whole range of x given.
  • y = mx + c: m is the gradient, c is where the line crosses the y-axis.
  • Gradient 0 gives a horizontal line y = c.
  • A sketch needs the direction of slope and the sign of the y-intercept.
  • Extended: rearrange to y = mx + c first, for example 3x + 2y = 5. A line x = k is vertical.

End-of-chapter exercise

Test yourself on the whole chapter. Work through these before moving on.

  1. Write down the gradient and the y-intercept of the line y = 2x - 3.2 marks
  2. Do not use a calculator. Complete the table of values for y = 4x - 3. | x | -1 | 0 | 3 | | --- | --- | --- | --- | | y | | | |3 marks
  3. Do not use a calculator. A line has equation y = -2x + 5. Find the coordinates of the points on the line where x = 0 and where x = 4. Explain how to finish drawing the line.3 marks
  4. A student draws the line y = 0.25x + 2 for -4 ≤ x ≤ 8. Find the y-values at the two ends of the line.2 marks
  5. Do not use a calculator. A line has gradient 0 and passes through the point (3, -2). Write down the equation of the line and the coordinates of the point where it crosses the y-axis.2 marks
  6. A sketch of y = -3x + 6 is needed. State the direction of slope of the line and where it crosses the y-axis.2 marks
  7. Write down the gradient and the y-intercept of the line y = 5 - 3x.2 marks
  8. Do not use a calculator. Rearrange 2x - 5y = 10 to make y the subject. Write down the gradient and the y-intercept.3 marks
  9. Do not use a calculator. A grid has x from -2 to 6 and y from -4 to 8. The line 2y - 3x = 4 is ruled across the grid as far as the grid allows. (a) Rearrange the equation to make y the subject. (b) Find the coordinates of the points where the line crosses the y-axis and the x-axis. (c) Find the coordinates of the two points where the line meets the edge of the grid.6 marks
  10. A taxi fare of F dollars for a journey of d km satisfies 5F - 8d = 18. The graph is drawn with d on the horizontal axis, from 0 to 20, and F on the vertical axis. (a) Rearrange the equation to make F the subject. (b) Find the coordinates of the two ends of the fare line. (c) The lines F = 30 and d = 5 are also drawn. State which of these two lines is horizontal and which is vertical. Find the coordinates of the point where each one meets the fare line.8 marks

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