Let us now draw the graphs of y = 2x − 1, y = 2x + 1, y = 2x + 5, first individually (as shown in the figure) and then on the same axes (as shown in the figure).
Step-by-step solution
Idea: Each equation needs two points. All three have the same coefficient of x (2), so they rise at the same rate; only the constant term changes.
- Points for each line:1 mark
Line x = 0 x = 2 x = −2 y = 2x − 1 (0, −1) (2, 3) (−2, −5) y = 2x + 1 (0, 1) (2, 5) (−2, −3) y = 2x + 5 (0, 5) (2, 9) (−2, 1) - Draw each line separately: plot two of its points, join them and extend the line.1 mark
- Now draw all three on the same axes (see the graph).1 mark
- The three lines never meet: they are parallel, because all have the same slope 2. They cut the y-axis at (0, −1), (0, 1) and (0, 5): changing the constant shifts the line up or down.1 mark
Check: Moving 1 unit right, each line rises 2 units: (0, −1) → (1, 1), (0, 1) → (1, 3), (0, 5) → (1, 7) ✓. Their vertical gaps stay the same everywhere (2 and 4 units), as parallel lines should.
Answer to write in the exam
y = 2x − 1: points (0, −1), (2, 3)
y = 2x + 1: points (0, 1), (2, 5)
y = 2x + 5: points (0, 5), (−2, 1)
Plot each line separately, then all three on the same axes (graph)
∴ The three lines are parallel (slope 2) and cut the y-axis at (0, −1), (0, 1), (0, 5)
Common mistakes that cost marks
- Plotting y = 2x − 1 through (0, 1). The constant term −1 gives (0, −1).
- Drawing the lines with different steepness. They all rise 2 units per unit to the right.
- Joining points from different lines, for example (0, −1) with (2, 5).
How this can come in the exam
Which line is parallel to y = 2x + 5?
- y = 5x + 2
- y = −2x + 5
- y = 2x − 7
- y = 12x + 5
Show answer
(C) y = 2x − 7
Parallel lines have the same slope; only y = 2x − 7 has slope 2.
Draw y = x + 2 and y = x − 3 on the same axes. Are they parallel? Where does each cut the y-axis?
Show answer
Points (0, 2), (2, 4) and (0, −3), (3, 0) (1 mark). Graph (1 mark). Parallel (both slope 1); y-axis at (0, 2) and (0, −3) (1 mark).Try one yourself
Write the equations of two lines parallel to y = 2x − 1, one passing through (0, 4) and one through (0, −6).
Show answer
y = 2x + 4 and y = 2x − 6.
More questions like this
- Does this help you to conclude anything about the linear equation y = ax + b when a is fixed but b varies?
(Hint: In these equations a = 2, and b takes the values −1, 1 and 5, respectively.) - Now let us draw the graphs of the equations y = x + 3, y = 2x + 5 and y = 3x − 2. See the figure and observe where these lines cut the y-axis.
- Draw the graphs of the following sets of lines. In each case, reflect on the role of ‘a’ and ‘b’.
- Write a polynomial of degree 3 in the variable x, in which the coefficient of the x2 term is −7.
- Find the values of the following polynomials at the indicated values of the variables.