All three forms represent a straight line, but each emphasizes different information. Gradient-intercept form shows the gradient and vertical intercept directly, general form places all terms on one side, and point-gradient form uses a known point and gradient.
This is SL 2.1, Equations of a line in IB Mathematics: Applications and Interpretation.
| Form | Standard structure | Main advantage |
|---|---|---|
| Gradient-intercept | Shows gradient and -intercept immediately |
| General | Convenient for rearranging equations and representing vertical lines |
| Point-gradient | Builds a line directly from gradient and point |
To convert from general form to gradient-intercept form, isolate . If , then , so the gradient is and the vertical intercept is , provided . If , the line is vertical. This makes general form useful for recognizing both ordinary and vertical lines.
For example, suppose a line has gradient and passes through . Begin with point-gradient form:
Expanding and making the subject gives gradient-intercept form:
Moving every term to one side gives general form:
These equations describe exactly the same line. General form is not unique because multiplying the entire equation by any non-zero constant gives an equivalent equation; for example, represents the same line.
A useful special case is a vertical line such as . It can be written in general form as , but not as , because a vertical line has an undefined gradient.
Common misconception: changing the form changes the line. It does not, providedprovided every algebraic operation is applied correctly to the whole equation.
For an IB exam response, choose the form that matches the given information. Use point-gradient form when given a point and gradient, then rearrange if the command asks for gradient-intercept or general form. Examiners expect correct algebra and the requested final form; a GDC is permitted on every Math AI paper, although these conversions are usually completed algebraically.
