Worksheet: Solving Cubic Equations Using Function Graphs

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In this worksheet, we will practice solving a cubic equation graphically.

Q1:

Use technology to plot the graph of 𝑓(𝑥)=𝑥+5𝑥−100, and use this graph to find the solutions to the equation 𝑥+5𝑥=100 to two decimal places.

  • A 𝑥 = 3 . 4 4
  • B 𝑥 = − 3 . 4 4
  • C 𝑥 = − 1 2 . 8 1
  • D 𝑥 = 7 . 0 3
  • E 𝑥 = − 7 . 0 3

Q2:

Use technology to plot the graph of 𝑓(𝑥)=𝑥+3𝑥−2, and use this graph to find the solutions to the equation 𝑥+3𝑥=2 to two decimal places.

  • A 𝑥 = − 0 . 7 3 , 𝑥 = 1 , 𝑥 = 2 . 7 3
  • B 𝑥 = − 2 . 7 3 , 𝑥 = − 1 , 𝑥 = 0 . 7 3
  • C 𝑥 = − 2 . 0 0 , 𝑥 = 1 . 0 0 , 𝑥 = 0 . 0 0
  • D 𝑥 = − 0 . 5 6 , 𝑥 = 0 , 𝑥 = 3 . 5 6
  • E 𝑥 = − 3 . 5 6 , 𝑥 = 0 , 𝑥 = 0 . 5 6

Q3:

The figure shows the graphs of the curve 𝑦=𝑥(𝑥−1)(𝑥+2) and the line 𝑦=𝑘(𝑥−1) for some 𝑘<0.

The graphs intersect at three points, one of which has 𝑥-coordinate 1. Find a quadratic equation whose roots are the 𝑥-coordinates of the other two points of intersection.

  • A 𝑥 ( 𝑥 + 2 ) + 𝑘 = 0
  • B 𝑥 ( 𝑥 + 2 ) − 𝑘 ( 𝑥 − 1 ) = 0
  • C 𝑥 ( 𝑥 − 1 ) = 𝑘
  • D 𝑥 ( 𝑥 + 2 ) − 𝑘 = 0
  • E 𝑥 ( 𝑥 + 2 ) = 0

By considering the discriminant of this quadratic equation, find the slope of the line through (1,0) which is tangent to the curve at some other point.

At what point is this line tangent to the curve?

  • A ( 3 , 3 0 )
  • B ( 0 , 0 )
  • C ( 1 , 0 )
  • D ( − 1 , 2 )
  • E ( 2 , 8 )

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