Solving Equations using graphs

Solving Equations using graphs

Understanding the Application of Graphs to Solve Equations

Basics of Solving Equations via Graphs

• When you solve an equation graphically, you are looking for the value of ‘x’ which makes the equation true.
• The values of ‘x’ that give you a true statement for an equation are its roots or solutions.
• If you plot the equation on a graph, these solutions show up as the points where your graph crosses the x-axis.
• In other words, they are the x-coordinates where the y-coordinate is equal to zero.

Intersection Points

• Another common use of graphs to solve equations is to find intersection points of two graphs.
• The x-values at the intersection points are the solutions to the equation when both the functions are set as equal to each other.
• If the two graphs intersect, it means they share a common pair of x and y coordinates - this is the solution to the resulting equation.

Quadratic Equations

• A quadratic graph, or parabola, may have 0, 1 or 2 real roots. This is determined by where it crosses the x-axis.
• If the quadratic crosses the x-axis twice, there are two distinct real roots.
• If it only touches the x-axis (at the vertex), there is one real root.
• A parabola with no x-intercepts has no real roots but will have two complex roots.

Practical Steps

• Firstly, rearrange the equation to give ‘y =’ form, then plot the graph.
• Identify where it crosses the x-axis to locate the roots.
• Equally, if using two equations, find where they intersect to obtain shared solutions.
• Use accurate drawing and read off values carefully to ensure solutions are correct.

General Caution

• Remember that a graph only provides an approximate solution, unless you use a method to find the exact intersection or root points.
• Always pair this method with an algebraic check to confirm the roots.
• Understanding the behaviour of graphs will help interpret whether the number of solutions found makes sense with the type of graph you have.