Study Notes
Linear graphs involve understanding and interpreting information from graphs, plotting points, and determining equations of lines. Key concepts include Cartesian coordinates, gradients, midpoints, and line equations.
- Cartesian Plane — a two-dimensional plane formed by two perpendicular number lines intersecting at zero. Example: The x-axis and y-axis form the Cartesian plane.
- Gradient — the measure of the steepness of a line, calculated as the change in y divided by the change in x. Example: For points A(3,5) and B(6,7), the gradient is (7-5)/(6-3) = 2/3.
- Midpoint — the point exactly halfway between two points on a line segment. Example: The midpoint of A(5,2) and B(1,-1) is ((5+1)/2, (2-1)/2) = (3, 0.5).
- Length of a Line Segment — the distance between two points, calculated using the Pythagorean theorem. Example: The length between A(5,2) and B(1,-1) is √((1-5)² + (-1-2)²) = √25 = 5.
- Parallel Lines — lines with equal gradients. Example: Lines with gradients 2 and 2 are parallel.
- Perpendicular Lines — lines whose gradients multiply to -1. Example: Lines with gradients 2 and -1/2 are perpendicular.
- Equation of a Straight Line — expresses the relationship between x and y coordinates, typically in the form y = mx + c. Example: A line with gradient 2 and y-intercept 3 has the equation y = 2x + 3.
Exam Tips
Key Definitions to Remember
- Cartesian Plane: A two-dimensional plane with perpendicular axes.
- Gradient: The steepness of a line, calculated as rise over run.
- Midpoint: The point halfway between two points on a line segment.
- Equation of a Line: y = mx + c, where m is the gradient and c is the y-intercept.
Common Confusions
- Confusing the x-intercept and y-intercept.
- Miscalculating the gradient by mixing up the change in x and y.
Typical Exam Questions
- What is the gradient of the line passing through (2,3) and (4,7)? Answer: (7-3)/(4-2) = 2
- Find the midpoint of the line segment joining (1,2) and (3,4). Answer: ((1+3)/2, (2+4)/2) = (2, 3)
- Write the equation of a line with gradient 3 and y-intercept -2. Answer: y = 3x - 2
What Examiners Usually Test
- Ability to calculate gradients and midpoints.
- Understanding of line equations and how to derive them.
- Recognition of parallel and perpendicular lines based on gradients.