QUESTION 1
iv
Easy
Draw the graphs of the following sets of lines. In each case, reflect on the role of 'a' and 'b'.
y = 3x − 1, y = 3x, y = 3x +1
SOLUTION
1
Recall equation of a straight line
A straight line is written in the form \(y = ax + b\), where \(a\) is its slope and \(b\) is the \(y\)- intercept.
2
Identify two points on each line
For \(y = 3x − 1\):
When \(x = 1, y = 2\)
When \(x = −1, y = −4\)
So the two points are (\(1,2\)) and (\(−1−4\))
For \(y = 3x\):
When \(x = 1, y = 3\)
When \(x = −1, y = −3\)
So the two points are (\(1,3\)) and (\(−1,−3\))
For \(y = 3x + 1\):
When \(x = 1, y = 4\)
When \(x = −1, y = −2\)
So the two points are (\(1,4\)) and (\(−1,−2\))
When \(x = 1, y = 2\)
When \(x = −1, y = −4\)
So the two points are (\(1,2\)) and (\(−1−4\))
For \(y = 3x\):
When \(x = 1, y = 3\)
When \(x = −1, y = −3\)
So the two points are (\(1,3\)) and (\(−1,−3\))
For \(y = 3x + 1\):
When \(x = 1, y = 4\)
When \(x = −1, y = −2\)
So the two points are (\(1,4\)) and (\(−1,−2\))
3
Observe the role of '\(a\)' and '\(b\)'
All three lines have the same slope, \(a = 3\), so they are parallel to one another.
While the slopes are same, they have different \(y-intercepts (b): −1, 0,\) and \( 1\), so the lines move vertically without changing tilt.
While the slopes are same, they have different \(y-intercepts (b): −1, 0,\) and \( 1\), so the lines move vertically without changing tilt.
Concept Note
When slope \(a > 1\), the line is steeper than the line \(y = x\), which is equally inclined to both axes. However, when \(a < 1\), the line is less steep than the line \(y = x\).
Changing \(b\) shifts the line vertically without changing slope.