Worksheet: Extrema of a Function Graph

In this worksheet, we will practice interpreting the minimum and maximum values and the end behavior of a function graph.


Find the maximum or minimum value of the function 𝑓(𝑥)=1+3𝑥, given 𝑥[3,3].

  • AThe maximum value is 0.
  • BThe minimum value is 0.
  • CThe minimum value is 3.
  • DThe minimum value is 1.
  • EThe maximum value is 1.


Consider the graph of the quartic shown.

Which of the points 𝐴, 𝐵, and 𝐶 is a local maximum?

  • A𝐶
  • B𝐵
  • C𝐴

Which of the points 𝐴, 𝐵, and 𝐶 is a local minimum?

  • A𝐴 and 𝐵
  • B𝐵 and 𝐶
  • C𝐴 and 𝐶

Which of the points 𝐴, 𝐵, and 𝐶 is a global minimum?

  • A𝐶
  • B𝐴
  • C𝐵

By considering the end behavior of the quartic, determine whether the leading coefficient is positive or negative.

  • APositive
  • BNegative


Consider the cubic graph shown.

What are the coordinates of the local maximum?

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

What are the coordinates of the local minimum?

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

If we consider the end behavior of this cubic, it enters into the bottom left quadrant and exits from the top right quadrant. Is the leading coefficient of this cubic positive or negative?

  • APositive
  • BNegative


What is the maximum value of the function 𝑦=5|2𝑥|?


Which of the following has the lowest minimum value?

  • Aa quadratic function whose graph cuts the 𝑥-axis at −1 and 2 and cuts the 𝑦-axis at −4
  • B
  • C𝑓(𝑥)=𝑥4𝑥
  • D𝑓(𝑥)=5(𝑥3)+4
  • E𝑘(𝑥)=13(𝑥3)(𝑥+4)


The following is the graph of 𝑓(𝑥)=6𝑥+8𝑥+1 showing the intersection with a line 𝑦=𝑐.

Solve 𝑓(𝑥)=45.

  • A𝑥=1,𝑥=4
  • B𝑥=2,𝑥=112
  • C𝑥=3,𝑥=13
  • D𝑥=2,𝑥=1
  • E𝑥=2,𝑥=14

Solve 𝑓(𝑥)=1.

  • A𝑥=2
  • B𝑥=7
  • C𝑥=3
  • D𝑥=3
  • E𝑥=5

The solution to the previous part shows that 1 is the minimum value of this function. Use the same idea to find the maximum value and the input that achieves this.

  • Amaximum value = 3 at 𝑥=7
  • Bmaximum value = 9 at 𝑥=13
  • Cmaximum value = 7 at 𝑥=2
  • Dmaximum value = 5 at 𝑥=12
  • Emaximum value = 13 at 𝑥=1

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