Coefficient a as a =tgα
Coefficient a describes the slope of the line. Geometrically, it is the tangent of angle α, which the line forms with the positive part of the X-axis.
a =tgα, so the larger the value of a, the steeper the line.
Linear function
Linear function formula
The formula y = ax + b tells you more than where the line lies. It also shows its slope, the angle with the X-axis, and the starting point on the Y-axis.
a = tan α
a = Δy / Δx
b - Y-axis intercept
Check your knowledge
A short quiz will help you quickly spot which concepts are worth reviewing.
a
slope
rate of change of y
b
starting point
where the line crosses the Y-axis
formula
y = ax + b
Coefficient a as an increment
The same coefficient can be calculated from two points on the line. You compare how much y changed and how much x changed.
a =ΔyΔx. For points (0, 2) and (1, 5), we have a = 3 / 1 = 3.
One line, one angle
Each value of coefficient a corresponds to a different angle of inclination. As angle α increases, the tangent value also increases, which is the slope coefficient of the line.
a = tg α connects the function formula with the geometry of the graph.
What does b mean?
The constant term b is the value of the function for x = 0. On the graph, it is the point where the line crosses the Y-axis.
In the function y = 3x - 1 the number b is -1, so the line passes through the point (0, -1).
How do you read the formula quickly?
First look at b to mark the point on the Y-axis. Then use a as a step: move 1 to the right and a up or down.
For y = 3x - 1 you start at (0, -1), then move 1 to the right and 3 up.
How does the sign of a change the graph?
The sign of the slope coefficient immediately tells you whether the line goes up, goes down, or stays horizontal.
a > 0
the function is increasing
a = 0
the function is constant
a < 0
the function is decreasing
Mini example with parameters
For the function f(x) = -2x + 5 we have a = -2 and b = 5. This means the line crosses the Y-axis at (0, 5), and when x increases by 1, the value of the function decreases by 2.
See how to draw the graph