Solving Equations Graphically (AQA A-Level Mathematics): Revision Notes

Steps to Solve Equations Graphically

1. Understand the Equation:

• Start with an equation that you want to solve, such as $f(x) = g(x)$.
• The solution(s) to this equation correspond to the x-coordinates where the graphs of $\ y = f(x)$ and $\ y = g(x)$ intersect.

2. Graph the Functions:

• Graph each function separately on the same coordinate plane.
• If the equation is $\ f(x) = 0$, graph $\ y = f(x)$ and find where it crosses the x-axis.
• For more complex equations, rearrange them if necessary to isolate the functions on either side.

3. Identify the Intersection Points:

• The points where the two graphs intersect represent the solutions to the equation.
• The x-coordinates of these intersection points are the solutions.

4. Estimate or Calculate the Solutions:

• If the graphs intersect at specific points, those x-values are the exact solutions.
• If the intersection occurs between grid points, you may need to estimate the solution or use technology (like a graphing calculator) for more accuracy.

Example 1: Solving a Linear-Quadratic Equation Solve $\ x^2 + x - 6 = 2x$ graphically.

Steps:

Rewrite the Equation:

• Move all terms to one side: $\ x^2 - x - 6 = 0 .$

• This is equivalent to finding where the quadratic $\ y = x^2 - x - 6$ intersects the line $\ y = 2x .$ Graph the Functions:

• Graph $\ y = x^2 - x - 6$ (a parabola) and $\ y = 2x$ (a straight line) on the same axes. Find the Intersection Points:

• Identify where the parabola and the line intersect. Suppose the graphs intersect at$\ x = -2$ and $\ x = 3 .$ Solution:

• The solutions are x = -2 and x = 3.

  

Example 2: Solving a Nonlinear Equation

Solve$\ x^3 - 4x = x + 2$ graphically.

Steps:

5. Rewrite the Equation:

• Rearrange to $\ x^3 - 4x - x - 2 = 0$, which simplifies to$\ x^3 - 5x - 2 = 0 .$
• The equation $\ x^3 - 5x - 2 = 0$ can be solved by graphing $\ y = x^3 - 5x - 2$.

6. Graph the Function:

• Plot the cubic function $\ y = x^3 - 5x - 2$ on a graph.
• Alternatively, you could also graph $\ y = x + 2$ and find their intersection with $\ y = x^3 - 4x .$

7. Identify the Intersection Points:

• Locate the point(s) where the graph crosses the x-axis. If it crosses at$\ ( x = -2), \ ( x = 1 )$, and $\ ( x = 2 )$, these are the solutions.

8. Solution:

• The solutions are x = -2, x = 1, and x = 2.