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.)
Step-by-step solution
Idea: On y = ax, moving 1 unit to the right raises y by a units. So a measures how steep the line is.
- All the lines pass through the origin: putting x = 0 in y = ax gives y = 0 for every a.1 mark
- When a > 0 the line rises from left to right. As a increases (⅓, ½, 1, 2, 3), the line becomes steeper: for each 1 unit to the right it goes up a units.1 mark
- y = x (a = 1) is equally inclined to both axes. If a > 1 (like 2x, 3x) the line is steeper than y = x; if a < 1 (like ½x, ⅓x) it is less steep than y = x. The number a is the slope of the line y = ax.1 mark
Check: At x = 1 the lines reach heights ⅓, ½, 1, 2, 3, in the same order as their steepness ✓.
Answer to write in the exam
x = 0 ⇒ y = a(0) = 0, so every line y = ax passes through (0, 0)
a > 0: line rises from left to right; larger a ⇒ steeper line
a > 1 (e.g. y = 2x, 3x): steeper than y = x
a < 1 (e.g. y = ½x, ⅓x): less steep than y = x
∴ a is the slope of the line y = ax
Common mistakes that cost marks
- Thinking a bigger a moves the line up (shifts it). For y = ax, changing a turns the line about the origin; it does not shift it.
- Saying y = ½x is steeper because ½ looks “smaller” than x. Compare the values of a: ½ < 1, so it is flatter.
- Forgetting that all these lines pass through (0, 0).
How this can come in the exam
Which of these lines is the steepest?
- y = 34x
- y = x
- y = 52x
- y = 2x
Show answer
(C) y = 52x
The largest slope is 52 = 2.5.
Assertion (A): The line y = 0.8x is less steep than the line y = x.
Reason (R): For y = ax with 0 < a < 1, the line is less steep than y = x.
- Both A and R are true, and R is the correct explanation of A.
- Both A and R are true, but R is not the correct explanation of A.
- A is true but R is false.
- A is false but R is true.
Show answer
(A) Both A and R are true, and R is the correct explanation of A.
0.8 < 1, so by R the line is flatter than y = x; R explains A.
Try one yourself
Arrange y = 4x, y = 14x, y = x and y = 1.5x from least steep to steepest.
Show answer
y = ¼x, y = x, y = 1.5x, y = 4x.
More questions like this
- 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).
- 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.)