Number of intersections between graphs

Just like we can find the number of roots of an equation, we can also find the number of intersections between two graphs.

Steps

When we have two equations, we can find the number of intersections between their graphs like this:

1. Write the equations in the form y=... by moving all terms to one side except the y term.
2. Set both equations equal to each other.
3. Write the resulting equation in the form ax^2 + bx + c = 0.
4. Calculate the discriminant using D = b^2 - 4ac.

A reminder of the meaning of the discriminant

• If D > 0, there are two intersections between the graphs.
• If D = 0, there is one intersection between the graphs (the graphs are tangent to each other).
• If D < 0, there are no intersections between the graphs.

Examples

Example: find the number of points where the graphs y = x^2 + 2x + 3 and y = 3x - 1 intersect

• No need to rearrange the equations, as they already have a y on one side on its own.
• Set the equations equal to each other:
  • x^2 + 2x + 3 = 3x - 1
• Rearrange to the form ax^2 + bx + c = 0:
  • x^2 + 2x + 3 - 3x + 1 = 0
  • x^2 - x + 4 = 0
• Calculate the discriminant:
  • D = (-1)^2 - 4 \times 1 \times 4 = -15
• Since D < 0, there are no intersections between the graphs.