Draw the graphs of y = 12x, y = x, y = 2x by selecting suitable points on these lines.
(Hint: In order to graph y = 12x, we could take the points (0, 0) and (4, 2). Can you verify that these lie on the line?)
Step-by-step solution
Idea: Two points fix a straight line. Choose x-values that make y a whole number (for y = ½x, take an even x).
- Verify the hint: for (0, 0): ½ × 0 = 0 ✓. For (4, 2): ½ × 4 = 2 ✓. So both points lie on y = ½x.1 mark
- Tables of points:1 mark
Line Point 1 Point 2 Extra check y = ½x (0, 0) (4, 2) (−4, −2) y = x (0, 0) (2, 2) (−3, −3) y = 2x (0, 0) (1, 2) (−1, −2) - Plot each pair of points, join them with a ruler and extend the line in both directions (see the graph).1 mark
- All three lines pass through the origin (0, 0). y = 2x is the steepest, y = x makes equal angles with both axes, and y = ½x is the least steep.1 mark
Check: Each line rises by its coefficient for every 1 unit to the right: from (0, 0), y = 2x reaches (1, 2), y = x reaches (1, 1), y = ½x reaches (1, ½) ✓.
Answer to write in the exam
y = ½x: x = 0 ⇒ y = 0; x = 4 ⇒ y = ½(4) = 2. Points (0, 0), (4, 2)
y = x: points (0, 0), (2, 2)
y = 2x: points (0, 0), (1, 2)
Plot each pair of points and join with a straight line (graph)
∴ All three lines pass through (0, 0); y = 2x is the steepest, y = ½x the least steep
Common mistakes that cost marks
- Choosing x = 1 for y = ½x and then plotting y = 2 instead of ½. Pick even x-values to avoid fractions.
- Joining the two points but not extending the line beyond them. A graph of an equation is the whole line.
- Drawing y = ½x steeper than y = x. A coefficient less than 1 gives a flatter line.
How this can come in the exam
Which point lies on the line y = 12x?
- (2, 4)
- (6, 3)
- (3, 6)
- (1, 2)
Show answer
(B) (6, 3)
½ × 6 = 3, so (6, 3) satisfies the equation.
Draw the graphs of y = 3x and y = 13x on the same axes. Which is steeper? Where do they meet?
Show answer
Points: y = 3x through (0, 0), (1, 3); y = ⅓x through (0, 0), (3, 1) (1 mark). Correct graph (1 mark). y = 3x is steeper; they meet at the origin (1 mark).Try one yourself
Find two points to draw y = 32x and say whether it is steeper than y = x.
Show answer
(0, 0) and (2, 3). Yes, it is steeper because 32 > 1.
More questions like this
- The figure shows all the three graphs on the same axes. Does this help you to conclude anything about the linear equation y = ax, a > 0 as a varies? What happens when a > 1 and when a < 1?
(Hint: You may also plot the equations y = 3x and y = 13x on the same axes.) - Now let us draw the graphs of y = −13x, y = −x, y = −3x by selecting suitable points on these lines.
- The figure shows all the three graphs on the same axes. Does this help you to conclude anything about the linear equation y = − ax, a > 0, as a varies? What will happen when a > 1 and when a < 1?
- Differentiate between the graphs of the equations y = 3x + 1, and y = −3x + 1.
- 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).