Summary
Graphs in algebra involve plotting points and interpreting various types of graphs, such as linear and quadratic graphs, on a coordinate plane.
- Cartesian Plane — A two-dimensional plane with two perpendicular axes intersecting at zero. Example: The x-axis and y-axis on a graph.
- Plotting a Point — Representing data as a point using coordinates (x, y). Example: The point (3, 4) is 3 units along the x-axis and 4 units up the y-axis.
- Distance Between Two Points — The length between two points calculated using the Pythagorean theorem. Example: The distance between (1, 2) and (4, 6) is 5 units.
- Midpoint of a Line Segment — The point that divides a line segment into two equal parts. Example: The midpoint of the segment joining (1, 2) and (3, 4) is (2, 3).
- Gradient of a Line — The ratio of the change in y-coordinates to the change in x-coordinates. Example: The gradient of the line through (1, 2) and (3, 6) is 2.
- Parallel Lines — Lines that never meet, having the same gradient. Example: y = 2x + 1 and y = 2x - 3 are parallel.
- Perpendicular Lines — Lines that intersect at a right angle, with gradients that are negative reciprocals. Example: y = x and y = -x are perpendicular.
- Equation of a Straight Line — Typically written as y = mx + c, where m is the gradient and c is the y-intercept. Example: y = 3x + 2 is a straight line equation.
- Quadratic Graph — A graph of the form f(x) = ax^2 + bx + c, forming a parabola. Example: f(x) = x^2 - 4x + 4 is a quadratic equation.
Exam Tips
Key Definitions to Remember
- Cartesian Plane
- Gradient of a Line
- Equation of a Straight Line
- Quadratic Graph
Common Confusions
- Mixing up the x and y coordinates when plotting points
- Confusing the gradient with the y-intercept
- Misidentifying the direction of a parabola
Typical Exam Questions
- How do you find the gradient of a line through two points? Use the formula (change in y) / (change in x).
- What is the equation of a line parallel to y = 3x + 1? y = 3x + c, where c is any constant.
- How do you determine if two lines are perpendicular? Check if the product of their gradients is -1.
What Examiners Usually Test
- Ability to plot and interpret points and graphs
- Understanding of gradient and intercepts
- Recognition of linear and quadratic graph shapes