Algebra and Graphs
Algebra and Graphs
Algebra and Graphs tests symbolic manipulation and the link between equations and their graphs: expanding and factorising, linear and quadratic equations, simultaneous equations, inequalities, sequences, functions and graph interpretation.
Core rules
- Expand brackets by multiplying every term; factorise by taking out the highest common factor first.
- A quadratic ax² + bx + c = 0 can be solved by factorising or by x = [−b ± √(b² − 4ac)]/(2a).
- Solve simultaneous equations by elimination or substitution; the solution is the intersection point of the graphs.
- When multiplying or dividing an inequality by a negative number, reverse the inequality sign.
- For y = mx + c, m is the gradient and c the y-intercept. A quadratic graph is a parabola; its turning point and roots can be read from completed-square or factorised form.
Worked example
Solve 2x + 3y = 12 and x − y = 1. From the second equation x = y + 1. Substitute: 2(y + 1) + 3y = 12, so 5y + 2 = 12, y = 2. Then x = 3. Check: 2(3) + 3(2) = 12 and 3 − 2 = 1. The graphs meet at (3, 2).
Traps
- Losing a minus sign when substituting or expanding.
- Forgetting that a quadratic may have two solutions, one repeated solution or none in the real numbers.
- Plotting points without checking the scale on each axis.
- Treating a sketch as exact: read required values from the given equation or labelled coordinates.