Lesson Worksheet: Features of Cubic Graphs Mathematics

In this worksheet, we will practice identifying features of cubic graphs, such as intercepts, local minima, local maxima, and points of inflection, and studying rotational and reflective symmetry of a cubic graph.


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


Consider the graph.

Find the interval which contains all the values of 𝑥 that satisfy the inequality 𝑥+3𝑥6𝑥8<0.

  • A(4,1)(2,)
  • B(,2)
  • C(,4)(1,2)
  • D(,)
  • E(0,2)


The given figure shows the graph of 𝑓(𝑥)=𝑥4𝑥+1.

Use the graph to determine the number of solutions to the equation 𝑥=4𝑥1.

Use the graph to determine the intervals in which the solutions to 𝑥=4𝑥1 lie.

  • A0.5<𝑥<0, 0<𝑥<0.5, and 4<𝑥<5
  • B1<𝑥<0, 0<𝑥<1, and 3<𝑥<4
  • C0.5<𝑥<0, 0<𝑥<2, and 3<𝑥<4
  • D1<𝑥<0.5, 0.5<𝑥<0, and 3<𝑥<4
  • E1<𝑥<0, 0<𝑥<1, and 4<𝑥<5


Consider the graph of the function 𝑦=(𝑥+2)2.

Write down the coordinates of the point of symmetry of the graph, if it exists.

  • AItdoesnotexist.
  • B(2,2)
  • C(2,2)
  • D(0,0)
  • E(1,1)


True or False: A cubic graph has reflectional symmetry.

  • AFalse
  • BTrue


True or False: A cubic graph has rotational symmetry.

  • ATrue
  • BFalse


Find all 𝑥-intercepts of the given cubic graph.

  • A𝑥=0, 𝑥=1, 𝑥=1
  • B𝑥=1
  • C𝑥=0, 𝑥=1
  • D𝑥=0
  • E𝑥=0, 𝑥=1


Fill in the blank: A cubic graph always has rotational symmetry about the .

  • Aorigin
  • Binflection point
  • Clocal minima
  • D𝑥-axis
  • Elocal maxima


Find the point of rotational symmetry of the given graph.

  • A(5,0)
  • B(5,1)
  • C(6,2)
  • D(5,1)
  • E(1,5)


Consider the given graph.

What is the point of local minimum?

  • A(1,0)
  • B(1,4)
  • C(0,0)
  • D(2,0)
  • E(0,2)

What is the point of local maximum?

  • A(0,2)
  • B(1,4)
  • C(0,0)
  • D(1,0)
  • E(2,0)

This lesson includes 1 additional question and 36 additional question variations for subscribers.

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