Linear graphs and straight lines: gradients and intercepts
Algebra • Graphs
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Key concepts
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Definition of a linear function
A linear function produces a straight line when graphed in the Cartesian plane. The algebraic form is y = mx + c for most non-vertical lines, where m denotes gradient and c denotes y-intercept. Linear functions have constant rate of change. Equal steps in x produce equal steps in y, so the graph has uniform slope and no curvature.
Gradient (slope) explained
Gradient measures the steepness and direction of a straight line. Gradient equals rise over run: m = (change in y) ÷ (change in x) = (y2 - y1)/(x2 - x1) when two distinct points (x1,y1) and (x2,y2) lie on the line. Positive gradient indicates an increasing function, negative gradient indicates a decreasing function, zero gradient indicates a horizontal line, and undefined gradient indicates a vertical line that cannot be written as y = mx + c.
Intercepts of a line
The y-intercept is the point where the line crosses the y-axis, found by setting x = 0 and reading c in y = mx + c. The x-intercept is the point where the line crosses the x-axis, found by setting y = 0 and solving mx + c = 0. Intercepts provide two exact points for sketching a line. If both intercepts are known, the line can be drawn precisely by joining those intercept points.
Sketching linear graphs
Sketching uses intercepts, gradient, or a table of values. Sketching from y = mx + c uses c as the starting point on the y-axis and then applies the gradient m as rise over run to draw the line. When only two points are known, plotting both points and joining them produces the straight-line graph. Labels for axes, scale markings, and arrowheads indicate continuation of the line.
Plotting equations that correspond to straight lines
Equations of the form y = mx + c represent straight-line graphs except for vertical lines x = a. Plotting uses either the y-intercept and gradient or a small table of x and y values to produce accurate points. Linear equations in other forms, such as ax + by + d = 0, convert to y = mx + c by isolating y. Conversion reveals gradient and y-intercept for plotting and interpretation.
Parallel and perpendicular lines in y = mx + c
Lines are parallel when they share the same gradient and different intercepts, so parallel lines have equal m values in y = mx + c. Parallel lines never meet and maintain constant separation. Perpendicular lines have gradients that multiply to -1. If one line has gradient m, a perpendicular line has gradient -1/m when both lines are non-vertical and non-horizontal. Vertical and horizontal lines are perpendicular: x = a is perpendicular to y = b.
Finding the equation from two points
The gradient between two distinct points (x1,y1) and (x2,y2) is m = (y2 - y1)/(x2 - x1). Substitution of m and one point into y = mx + c yields c, then the full equation becomes y = mx + c. Algebraic rearrangement provides the final form. If points produce a vertical line (same x-value), the equation is x = constant and not expressible as y = mx + c.
Finding the equation from a point and a gradient
Given a point (x1,y1) and gradient m, substitution into y = mx + c gives y1 = m x1 + c. Solving for c produces c = y1 - m x1 and the line equation y = mx + (y1 - m x1). This method applies to any non-vertical line. For a vertical line, the gradient is undefined and the equation has the form x = constant.
Key notes
Important points to keep in mind