SQA Higher Maths Solving equations graphically
This page covers the following topics:
1. Equations of a straight line
2. Quadratic equations from graphs
3. Solutions of equations from graphs
4. Graphs of complex equations
A straight line graph has an equation of y = mx + c, where m is the gradient of the graph and c is the y-intercept of the graph, ie. the point at which the line crosses the y-axis. The gradient of the line can be calculated using: m = change in y/change in x. Another equation that can be used to find the equation of a straight line is y − y₁ = m(x − x₁), where m is the gradient of the graph and the graph passes through the point (x₁, y₁).
A quadratic equation is written in the form y = ax² + bx + c. To find the equation of a quadratic from its graph, the values of a, b and c must be calculated. To do so, three points on the graph must be known. These can be substituted in the general form of a quadratic, which would give 3 equations that can be solved simultaneously to find the values of a, b and c.
Systems of simultaneous equations can be solved by graphing: the solutions to the system will be the points of intersection on their graphs.
Graphs that involve negative quadratic equations and cubic equations can be solved just as linear and quadratic equations. The typical shapes of a negative quadratic and cubic graphs are given in the diagram.
The graphs of two equations are drawn on the same set of axes. Given that the two graphs intersect twice, state the number of solutions for this pair of simultaneous equations and explain your answer.
Graphically determine the number of solutions to the system of simultaneous equations: y + x = 8, y = x² + 12x + 20.
Find the equation of the graph given in the diagram.
Find the equation of the straight line in the diagram, given that it passes through (−8, 2) and (1, −10).
The graphs of a cubic equation and a linear equation are drawn on the same set of axes. Given that the two graphs do not intersect, state the number of solutions for this pair of simultaneous equations.
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