Draw the graphs of the following sets of lines. In each case, reflect on the role of 'a' and 'b'. (i) y = 4x, y = 2x, y = x (ii) y = −6x, y = −3x, y = −x (iii) y = 5x, y = −5x (iv) y = 3x − 1, y = 3x, y = 3x + 1 (v) y = −2x − 3, y = −2x, y = 2x + 3
A straight line is completely determined by any two distinct points on it. For each equation, we choose the two convenient values x = 0 and x = 1 to find the coordinates to plot.
Points to plot (using x = 0 and x = 1):
| Equation | When x = 0 | When x = 1 |
|---|---|---|
| y = 4x | (0, 0) | (1, 4) |
| y = 2x | (0, 0) | (1, 2) |
| y = x | (0, 0) | (1, 1) |
Reflection on a and b:
All three equations have b = 0, so all three lines pass through the origin (0, 0).
The values of a are 4, 2, and 1. All are positive, so all three lines rise from left to right.
As the value of a increases, the line tilts more sharply upwards and becomes steeper.
Points to plot (using x = 0 and x = 1):
| Equation | When x = 0 | When x = 1 |
|---|---|---|
| y = −6x | (0, 0) | (1, −6) |
| y = −3x | (0, 0) | (1, −3) |
| y = −x | (0, 0) | (1, −1) |
Reflection on a and b:
For all three lines, b = 0, so all pass through the origin (0, 0).
Their a-values are −6, −3, and −1. Since a is negative, all three lines fall from left to right.
Comparing their absolute magnitudes: |−6| > |−3| > |−1|, meaning y = −6x is the steepest and y = −x is the least steep.
Points to plot (using x = 0 and x = 1):
| Equation | When x = 0 | When x = 1 |
|---|---|---|
| y = 5x | (0, 0) | (1, 5) |
| y = −5x | (0, 0) | (1, −5) |
Reflection on a and b:
Both lines have b = 0, so both pass through the origin (0, 0).
Their a-values are 5 and −5. They have the same absolute magnitude:
The two lines form symmetric mirror images of each other. Changing the sign of a reverses the direction of the line without changing its steepness.
Points to plot (using x = 0 and x = 1):
| Equation | When x = 0 | When x = 1 |
|---|---|---|
| y = 3x − 1 | (0, −1) | (1, 2) |
| y = 3x | (0, 0) | (1, 3) |
| y = 3x + 1 | (0, 1) | (1, 4) |
Reflection on a and b:
All three equations have the same coefficient a = 3. Therefore, all three lines have identical steepness and slope, making them parallel.
Their b-values are −1, 0, and 1. When x = 0, y = b, so they intersect the y-axis at (0, −1), (0, 0), and (0, 1) respectively.
Points to plot (using x = 0 and x = 1):
| Equation | When x = 0 | When x = 1 |
|---|---|---|
| y = −2x − 3 | (0, −3) | (1, −5) |
| y = −2x | (0, 0) | (1, −2) |
| y = 2x + 3 | (0, 3) | (1, 5) |
Reflection on a and b:
For the first two lines, y = −2x − 3 and y = −2x, the value of a is the same (a = −2). Hence, these two lines have identical slope and are parallel. Their b-values are different (−3 and 0), so their y-intercepts are (0, −3) and (0, 0).
For the third line, y = 2x + 3, the value of a is 2 (positive), so it slopes upward from left to right with y-intercept (0, 3).
Although y = −2x and y = 2x + 3 have the same steepness (|−2| = |2| = 2), their directions are opposite because their signs of a differ.
For any linear relationship in standard form:
1. Role of a (Slope / Steepness / Direction):
2. Role of b (y-Intercept / Vertical Position):