QUESTION 1
i
Easy
Draw the graphs of the following sets of lines. In each case, reflect on the role of 'a' and 'b'.
y = 4x, y = 2x, y = x
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. Here, \(b = 0\).
2
Identify two points on each line
For \(y = 4x\):
When \(x = 1, y = 4\)
When \(x = −1, y = −4\)
So the two points are (\(1,4\)) and (\(−1,−4\))
For \(y = 2x\):
When \(x = 1, y = 2\)
When \(x = −1, y = −2\)
So the two points are (\(1,2\)) and (\(−1,−2\))
For \(y = x\):
When \(x = 1, y = 1\)
When \(x = 2, y = 2\)
So the two points are (\(1,1\)) and (\(2,2\))
When \(x = 1, y = 4\)
When \(x = −1, y = −4\)
So the two points are (\(1,4\)) and (\(−1,−4\))
For \(y = 2x\):
When \(x = 1, y = 2\)
When \(x = −1, y = −2\)
So the two points are (\(1,2\)) and (\(−1,−2\))
For \(y = x\):
When \(x = 1, y = 1\)
When \(x = 2, y = 2\)
So the two points are (\(1,1\)) and (\(2,2\))
3
Observe the role of '\(a\)' and '\(b\)'
As slope(\(a\)) increases, the line gets steeper.
The three lines are of the form \(y = ax\), so \(b = 0\), which means they all pass through the origin(\(0,0\)).
The three lines are of the form \(y = ax\), so \(b = 0\), which means they all pass through the origin(\(0,0\)).
Concept Note
Straight lines representing an equation of the form \(y = ax\) always pass through the origin(\(0,0\)).
When \(a > 1\), 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 \(y = x\).