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.

Q1:

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.

Q2:

Consider the graph of the quartic shown.

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

  • A 𝐶
  • B 𝐵
  • C 𝐴

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

  • A 𝐴 and 𝐵
  • B 𝐵 and 𝐶
  • C 𝐴 and 𝐶

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

  • A 𝐶
  • B 𝐴
  • C 𝐵

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

  • APositive
  • BNegative

Q3:

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

Q4:

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

Q5:

Consider a function 𝑓(𝑥)=𝑎𝑥+𝑏, where 𝑎, 𝑏, and 𝑛 are integers larger than 1. Which of the following statements is true?

  • AThe existence of extremes cannot be determined without more information.
  • BIf 𝑛 is odd, there will be an absolute maximum.
  • CIf 𝑛 is even, there will be an absolute minimum.
  • DThere will be neither an absolute maximum nor an absolute minimum.

Q6:

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
    𝑥 0 1 2 3 4 5 6 7
    𝑔 ( 𝑥 ) 6 0 −4 −6 −6 −4 0 6
  • C 𝑓 ( 𝑥 ) = 𝑥 4 𝑥
  • D 𝑓 ( 𝑥 ) = 5 ( 𝑥 3 ) + 4
  • E 𝑘 ( 𝑥 ) = 1 3 ( 𝑥 3 ) ( 𝑥 + 4 )

Q7:

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

Solve 𝑓(𝑥)=45.

  • A 𝑥 = 1 , 𝑥 = 4
  • B 𝑥 = 2 , 𝑥 = 1 1 2
  • C 𝑥 = 3 , 𝑥 = 1 3
  • D 𝑥 = 2 , 𝑥 = 1
  • E 𝑥 = 2 , 𝑥 = 1 4

Solve 𝑓(𝑥)=1.

  • A 𝑥 = 2
  • B 𝑥 = 7
  • C 𝑥 = 3
  • D 𝑥 = 3
  • E 𝑥 = 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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