Worksheet: Solving Cubic Equations Graphically

In this worksheet, we will practice graphing a cubic equation and using this graph to find its solution or solutions.


Use technology to plot the graph of 𝑓(𝑥)=𝑥+5𝑥100, and use this graph to find the solutions to the equation 𝑥+5𝑥=100 to two decimal places.

  • A𝑥=3.44
  • B𝑥=3.44
  • C𝑥=12.81
  • D𝑥=7.03
  • E𝑥=7.03


Use technology to plot the graph of 𝑓(𝑥)=𝑥+3𝑥2, and use this graph to find the solutions to the equation 𝑥+3𝑥=2 to two decimal places.

  • A𝑥=0.73, 𝑥=1, 𝑥=2.73
  • B𝑥=2.73, 𝑥=1, 𝑥=0.73
  • C𝑥=2.00, 𝑥=1.00, 𝑥=0.00
  • D𝑥=0.56, 𝑥=0, 𝑥=3.56
  • E𝑥=3.56, 𝑥=0, 𝑥=0.56


The figure shows the graphs of the curve 𝑦=𝑥(𝑥1)(𝑥+2) and the line 𝑦=𝑘(𝑥1) 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.

  • A𝑥(𝑥+2)+𝑘=0
  • B𝑥(𝑥+2)𝑘(𝑥1)=0
  • C𝑥(𝑥1)=𝑘
  • D𝑥(𝑥+2)𝑘=0
  • E𝑥(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(3,30)
  • B(0,0)
  • C(1,0)
  • D(1,2)
  • E(2,8)


The graph of 𝑦=𝑥2𝑥 is shown in the figure. What line needs to be drawn to solve 1=𝑥3𝑥1?

  • A𝑦=𝑥+2
  • B𝑦=2
  • C𝑦=𝑥2
  • D𝑦=𝑥+1
  • E𝑦=𝑥1


If the graph of the function 𝑓(𝑥)=𝑥6𝑥+11𝑥6 is as shown, use this graph to find the solution set to the equation 𝑥6𝑥+11𝑥=6.

  • A{1,3}
  • B{1,2}
  • C{1}
  • D{1,2,3}
  • E{2.3}

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