Solving Equations using Graphs

How do we use graphs to solve equations?

• Solutions are always read off the x-axis
• Solutions of f(x) = 0 are where the graph of y = f(x) cuts the x-axis
• If given g(x)=0 instead (Q: "by drawing a suitable straight line") then:
  • Rearrange into f(x) = mx + c and draw the line y = mx + c
  • Solutions are the x-coordinates of where the line crosses the curve

Note:  Solutions may also be called roots

Example: The graph of y = x³ + - 3x - 1 is shown below.
Use the graph to estimate the solutions (to I dp.) of the equation x³ + x² - 4x = 0.

We are given a different equation to the one plotted so we must rearrange it to f(x) = mx + c.
x³ + x² - 4x = 0
Add x - 1 to both sides :
x³ + x² - 3x - 1 = x - 1
Now plot y = mx + c (which is y = x - 1 here) on the graph - this is the solid red line on the graph below.
The solutions are the x coordinates of the crossing points so :
x = - 2.6, x = 0, x = 1.6