Multiplicity — cross or bounce

What this lesson shows

Take f of x equals, bracket, x minus 2, squared, times, bracket, x plus 3. Its roots are 2 and minus 3. But the graph does something different at each one.

Try it yourself

Real questions from this lesson's worksheets — answers below each one.

1. For f(x) = (x − 1)²(x + 2), at x = 1 the graph ___ the x-axis.
   Show answer

   bounces (touches)
2. For f(x) = (x − 1)²(x + 1), at x = 1 the graph ___ the x-axis.
   Show answer

   bounces (touches)
3. Which function crosses the x-axis at x = 5 and x = −2?
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   (x − 5)(x + 2)
4. For f(x) = (x − 1)²(x + 3), at x = 1 the graph ___ the x-axis.
   Show answer

   bounces (touches)