Linear functions
- Author:
- P Porras
A function is said to be a linear function if there is only one variable with the exponent one:
where k is the gradient (the slope of the function) and c is a vertical intercept. The graph of a linear function is a straight line which is NOT parallel to x-axis ().
As it can be seen from the applet, the sign of the gradient defines if the linear graph is ascending or descending:
k > 0: the linear graph is ascending
k < 0: the linear graph is descending.
If one point and the gradient k are known, the linear function is uniquely defined. The linear function can be defined by the formula
Example 4. Define the linear function which is passing through the point (-2, 1) and has the gradient -3.
The known point (x0, y0) is (-2, 1) and the gradient k = -3. By substituting this to the formula, we get
The straight line between two points is unique and the linear function can be defined with two points. The gradient is the slope of the line: the greater gradient (without the sign) the steeper slope.
Line through two points
When defining the line with two points (x1, y1) and (x2, y2), Δy and Δx are obtained with the points:
Example 2. Define the line going through points (-2, 1) and (3, 5).
Solution 2. The known points are and . Thus, the slope is
The equation of the line is
The equation of the line can be also given is the standard form As a function, it would be written as
Example 3. The pressure at the beginning was 6.0 MPa and one minute later 0 MPa. If the pressure is linearly dependent on time, what was the pressure at t = 32 s.
Known points are ja
That is,
Using the equations as a function, we can substitute given time t = 32 s and the pressure seems to be
Drawing the linear function
1. If the point is given, use it as a starting point. If a function is known, use the intercept as a starting point.
2. Go to the right from
the starting point by the denominator of the gradient and up or down by the numerator of the gradient.
3. If the gradient is negative, go down and vice versa. This is your second point.Draw the linear graph through these points.