Linear functions – graphs and the equation ax + b = 0

What it shows

In y = ax + b the two coefficients play separate roles. The constant b is the y-intercept: changing it translates the line up or down without turning it. The slope a is the change in y when x increases by 1, which the slope triangle shows directly; a > 0 gives an increasing function, a < 0 a decreasing one, and a = 0 a horizontal line. The x-intercept is the solution x = −b/a of the linear equation ax + b = 0. Comparing the slopes and intercepts of two lines decides whether they intersect, are parallel or coincide.

How to use

Drag the point on the y-axis to change Coefficient b and the square point to change Coefficient a, with the slope triangle switched on. Ask students to predict where the line crosses the x-axis before reading the solution of ax + b = 0, and move point P along the line to read coordinates. Then turn on Second line and adjust Coefficient a' and Coefficient b' until the two lines are parallel, then coincident.

Parameters you can change

• Coefficient a -5–5
• Coefficient b -6–6
• Show slope triangle
• Show second line y = a'x + b'
• Coefficient a' -5–5
• Coefficient b' -6–6

Questions to explore

1. When a = 0, what does the graph look like, and does ax + b = 0 still have exactly one solution?
2. Where on the graph of y = 2x − 3 can you read the solution of 2x − 3 = 0?
3. Which values of a' and b' make the second line parallel to y = 2x + 1?