Worksheet: Cubic Equations

In this worksheet, we will practice writing a cubic equation given three x-intercepts and one point on the graph.

Q1:

The figure shows the curve 𝑦 = 𝑥 2 𝑥 3 together with the line 𝑦 = 𝑘 ( 𝑥 1 ) 1 which has slope 𝑘 and passes through point ( 1 , 1 ) .

Write a cubic polynomial whose roots are the numbers 𝑎 , 𝑏 , and 1.

  • A 𝑥 2 𝑥 3
  • B 𝑥 ( 𝑘 2 ) 𝑥 + 𝑘 + 1 3
  • C 𝑥 ( 𝑘 + 2 ) 𝑥 + 𝑘 1 3
  • D 𝑥 ( 𝑘 + 2 ) 𝑥 + 𝑘 + 1 3
  • E 𝑥 + ( 𝑘 + 2 ) 𝑥 + 𝑘 + 1 3

Divide this polynomial by 𝑥 1 to get a quadratic that is a multiple of ( 𝑥 𝑎 ) ( 𝑥 𝑏 ) .

  • A 𝑥 + 𝑥 𝑘 1 2
  • B 𝑥 + 𝑥 + 𝑘 + 1 2
  • C 𝑥 + 𝑥 𝑘 2
  • D 𝑥 + 𝑥 + 𝑘 1 2
  • E 𝑥 𝑥 𝑘 1 2

Since 𝑏 > 𝑎 , determine 𝑏 in terms of 𝑘 .

  • A 1 𝑘 + 1
  • B 1 4 𝑘 + 5 2
  • C 1 + 4 𝑘 + 5 2
  • D 𝑘 + 1
  • E 1 + 4 𝑘 + 1 2

Imagine changing the value of the slope 𝑘 so that the value of 𝑏 gets closer and closer to 1. When 𝑏 = 1 , the line will be tangent to the curve at the point ( 1 , 1 ) . Determine the equation of the tangent to the curve at the point ( 1 , 1 ) .

  • A 𝑦 = 𝑥 2
  • B 𝑦 = 5 𝑥 6
  • C 𝑦 = 𝑥
  • D 𝑦 = 𝑥 + 2
  • E 𝑦 = 3 𝑥 4

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