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Lesson: Writing a Cubic Equation Given Three X-Intercepts and One Point on the Graph

Worksheet • 1 Question

Q1:

The figure shows the curve 𝑦 = π‘₯ βˆ’ 2 π‘₯ 3 together with the line 𝑦 = π‘˜ ( π‘₯ βˆ’ 1 ) βˆ’ 1 which has slope π‘˜ and passes through point ( 1 , βˆ’ 1 ) .

Write a cubic polynomial whose roots are the numbers π‘Ž , 𝑏 , and 1.

  • A π‘₯ βˆ’ ( π‘˜ + 2 ) π‘₯ + π‘˜ + 1 3
  • B π‘₯ + ( π‘˜ + 2 ) π‘₯ + π‘˜ + 1 3
  • C π‘₯ βˆ’ ( π‘˜ βˆ’ 2 ) π‘₯ + π‘˜ + 1 3
  • D π‘₯ βˆ’ ( π‘˜ + 2 ) π‘₯ + π‘˜ βˆ’ 1 3
  • E π‘₯ βˆ’ 2 π‘₯ 3

Divide this polynomial by π‘₯ βˆ’ 1 to get a quadratic that is a multiple of ( π‘₯ βˆ’ π‘Ž ) ( π‘₯ βˆ’ 𝑏 ) .

  • A π‘₯ + π‘₯ βˆ’ π‘˜ βˆ’ 1 2
  • B π‘₯ βˆ’ π‘₯ βˆ’ π‘˜ βˆ’ 1 2
  • C π‘₯ + π‘₯ + π‘˜ βˆ’ 1 2
  • D π‘₯ + π‘₯ + π‘˜ + 1 2
  • E π‘₯ + π‘₯ βˆ’ π‘˜ 2

Since 𝑏 > π‘Ž , determine 𝑏 in terms of π‘˜ .

  • A βˆ’ 1 + √ 4 π‘˜ + 5 2
  • B 1 π‘˜ + 1
  • C βˆ’ 1 βˆ’ √ 4 π‘˜ + 5 2
  • D βˆ’ 1 + √ 4 π‘˜ + 1 2
  • E √ π‘˜ + 1

Imagine changing the value of the slope π‘˜ 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 ( βˆ’ 1 , 1 ) . Determine the equation of the tangent to the curve at the point ( βˆ’ 1 , 1 ) .

  • A 𝑦 = π‘₯ βˆ’ 2
  • B 𝑦 = 5 π‘₯ βˆ’ 6
  • C 𝑦 = π‘₯ + 2
  • D 𝑦 = π‘₯
  • E 𝑦 = 3 π‘₯ βˆ’ 4
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