Mathematics
Answer
Both equations are of the form y = ax + b.
To draw the graph, we identify suitable points on each line.
Equation 1 : y = 3x + 1
x = 1, y = 4
x = 2, y = 7
x = -1, y = -2
Equation 2 : y = –3x + 1
x = 0, y = 1
x = 2, y = -5
x = -2, y = 7
Similarities:
Both lines have the same y-intercept, b = 1, so both lines cut the y-axis at the same point (0, 1).
Differences:
The line y = 3x + 1 has a positive slope (a = 3), so it rises from the bottom-left to the top-right of the coordinate plane. It represents linear growth.
The line y = -3x + 1 has a negative slope (a = -3), so it falls from the top-left to the bottom-right of the coordinate plane. It represents linear decay.
Hence, the two lines pass through the same point (0, 1) on the y-axis but have slopes opposite in sign, so one rises and the other falls.
Related Questions
Consider the relationship between temperature measured in degrees Celsius (°C) and degrees Fahrenheit (°F), which is given by °C = a °F + b. Find a and b, given that ice melts at 0 degrees Celsius and 32 degrees Fahrenheit, and water boils at 100 degrees Celsius and 212 degrees Fahrenheit. (Hint: When °C = 0, °F = 32 and when °C = 100, °F = 212. Use this information to find a and b, and thus, the linear relationship between °C and °F.)
Identify other points on the line y = 2x + 1 by completing the following table.
x y 1 3 2 5 7 15 9 12 20 
Does this help you to conclude anything about the linear equation y = ax + b when a is fixed but b varies?
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