Learnify Academy is a tuition centre in Bahrain. Classes are for students in Bahrain only.Tuition classes in Bahrain only

Slope of a line · 3 marks

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?

xy−7−6−5−4−3−2−1123456−4−3−2−11230y = −3xy = −xy = −⅓x
Answer: Every line y = −ax (a > 0) passes through the origin and goes down from left to right (negative slope). The bigger a is, the steeper the line: for a > 1 it is steeper than y = −x, and for a < 1 it is less steep than y = −x.

Step-by-step solution

Given: The lines y = −⅓x, y = −x, y = −3x on the same axes

Idea: On y = −ax, moving 1 unit to the right lowers y by a units. The slope is −a, a negative number.

  1. All the lines pass through the origin: x = 0 gives y = 0.1 mark
  2. Because the slope −a is negative, each line falls from left to right: as x increases, y decreases (like linear decay). As a increases (⅓, 1, 3), the line falls more sharply.1 mark
  3. y = −x (a = 1) is equally inclined to both axes. If a > 1 (for example y = −3x) the line is steeper than y = −x; if a < 1 (for example y = −⅓x) it is less steep than y = −x.1 mark
Lines y = −ax (a > 0) pass through the origin and fall from left to right; when a > 1 they are steeper than y = −x, and when a < 1 they are less steep.

Check: At x = 1 the lines are at −⅓, −1 and −3: the further below the x-axis, the steeper the line ✓.

Answer to write in the exam

x = 0 ⇒ y = 0, so every line y = −ax passes through (0, 0)

Slope −a < 0: line falls from left to right; larger a ⇒ steeper line

a > 1 (e.g. y = −3x): steeper than y = −x

∴ a < 1 (e.g. y = −⅓x): less steep than y = −x

Common mistakes that cost marks

  • Saying y = −3x is less steep because −3 is “smaller” than −1. Steepness depends on the size of a, here 3 > 1.
  • Drawing these lines rising to the right. A negative slope means the line falls to the right.
  • Forgetting that the lines all pass through the origin.

How this can come in the exam

MCQ (1 mark)

Which line falls most steeply from left to right?

  1. y = −12x
  2. y = −x
  3. y = −4x
  4. y = 2x
Show answer

(C) y = −4x
Among the falling lines, −4x has the largest a = 4. (y = 2x rises.)

Assertion–Reason (1 mark)

Assertion (A): The graph of y = −5x represents linear decay.
Reason (R): A straight line with negative slope represents linear decay.

  1. Both A and R are true, and R is the correct explanation of A.
  2. Both A and R are true, but R is not the correct explanation of A.
  3. A is true but R is false.
  4. A is false but R is true.
Show answer

(A) Both A and R are true, and R is the correct explanation of A.
The slope of y = −5x is −5, which is negative, so by R it shows linear decay.

Try one yourself

Arrange y = −2x, y = −14x, y = −x from least steep to steepest.

Show answer

y = −¼x, y = −x, y = −2x.

More questions like this

All Linear polynomials questions · All maths questions

← Now let us draw the graphs of y = −1/3x, y = −x, y = −3x by selecting Differentiate between the graphs of the equations y = 3x + 1, and y = →