Lesson Worksheet: Solving Cubic Equations Graphically Mathematics

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

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.

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

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.

  • 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

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.

  • A𝑥(𝑥+2)+𝑘=0
  • B𝑥(𝑥+2)−𝑘(𝑥−1)=0
  • C𝑥(𝑥−1)=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)

Q4:

The graph of 𝑦=𝑥−2𝑥 is shown in the figure. What line needs to be drawn to solve 1=𝑥−3𝑥−1?

  • A𝑦=𝑥+2
  • B𝑦=2
  • C𝑦=𝑥−2
  • D𝑦=𝑥+1
  • E𝑦=𝑥−1

Q5:

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

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

Q6:

The given figure shows the graph of 𝑓(𝑥)=𝑥−4𝑥+1.

Use the graph to determine the number of solutions to the equation 𝑥=4𝑥−1.

Use the graph to determine the intervals in which the solutions to 𝑥=4𝑥−1 lie.

  • A−0.5<𝑥<0, 0<𝑥<0.5, and 4<𝑥<5
  • B−1<𝑥<0, 0<𝑥<1, and 3<𝑥<4
  • C−0.5<𝑥<0, 0<𝑥<2, and 3<𝑥<4
  • D−1<𝑥<−0.5, −0.5<𝑥<0, and 3<𝑥<4
  • E−1<𝑥<0, 0<𝑥<1, and 4<𝑥<5

Q7:

Use technology to plot the graph of 𝑓(𝑥)=−𝑥−𝑥−𝑥+1, and use this graph to choose the best solution to the equation 𝑥+𝑥+𝑥=1.

  • A𝑥=−1.039
  • B𝑥=0.544
  • C𝑥=−0.544
  • D𝑥=−1
  • E𝑥=1.039

Q8:

The graph of 𝑦=3𝑥+5𝑥−3𝑥−4 is shown in the figure. What is the equation of the line that needs to be drawn to solve 3𝑥+5𝑥−5𝑥=7?

  • A𝑦=2𝑥−3
  • B𝑦=3𝑥+2
  • C𝑦=−2𝑥−3
  • D𝑦=2𝑥+3
  • E𝑦=3𝑥−2

Q9:

Use technology to plot the graph of 𝑓(𝑥)=−2𝑥+5𝑥+5𝑥+2, and use this graph to find the number of real solutions to the equation 2𝑥−5𝑥−5𝑥=2.

Q10:

If the graph of the function 𝑓(𝑥)=12𝑥+𝑥−2𝑥−3 is as shown, use this graph to choose the best solution to the equation 𝑥+2𝑥=4𝑥+6.

  • A𝑥=1.866, 1.211, 2.655
  • B𝑥=−1.866, 1.211, 2.655
  • C𝑥=−1.866, −1.211, 2.655
  • D𝑥=1.866, −1.211, −2.655
  • E𝑥=−1.866, −1.211, −2.655

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