Math Problem Statement

For the function shown, determine the following.

the Period the Maximum the Minimum the Midline. Express as a line: the Amplitude Knowing that this is a periodic function, what would the output value be when ?

Solution

Let's analyze the graph to determine the required characteristics of the function:

  1. Period: The period is the distance along the x-axis after which the function starts repeating itself. Observing the graph, the function repeats its pattern approximately every 10 units along the x-axis. So, the period is 10.

  2. Maximum: The maximum value is the highest y-value the function reaches. From the graph, the maximum y-value is 5.

  3. Minimum: The minimum value is the lowest y-value the function reaches. The minimum y-value is -5.

  4. Midline: The midline is the horizontal line that represents the average of the maximum and minimum values. It is calculated as: Midline=Maximum+Minimum2=5+(5)2=02=0\text{Midline} = \frac{\text{Maximum} + \text{Minimum}}{2} = \frac{5 + (-5)}{2} = \frac{0}{2} = 0 So, the midline is the line y=0y = 0.

  5. Amplitude: The amplitude is the distance from the midline to either the maximum or minimum value. Since the maximum is 5 and the midline is 0, the amplitude is: Amplitude=MaximumMidline=50=5\text{Amplitude} = \text{Maximum} - \text{Midline} = 5 - 0 = 5

For the output value when x=?x = ?, please provide the specific x-value you are interested in.

Would you like more details on any of these calculations?

Here are some related questions:

  1. How do we calculate the period of a non-uniform periodic function?
  2. What are the implications of a shifted midline for amplitude calculations?
  3. How does the amplitude change if the midline is not at y=0y = 0?
  4. Can a function have multiple periods, and how would that affect the graph?
  5. How would you determine the equation of this function based on its graph?

Tip: When identifying the period from a graph, always ensure that you check multiple cycles to confirm consistency.

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Math Problem Analysis

Mathematical Concepts

Periodic Functions
Amplitude
Midline

Formulas

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Theorems

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Suitable Grade Level

Grades 9-12