Math Problem Statement

Find a function of the form y = A sin(kx) + C or y = A cos(kx) + C whose graph matches the function shown below. Leave your answer in exact form; if necessary, type pi for π.

Solution

We are asked to find a trigonometric function of the form y=Asin(kx)+Cy = A \sin(kx) + C or y=Acos(kx)+Cy = A \cos(kx) + C that matches the graph shown.

Steps to determine the equation:

  1. Amplitude AA: The amplitude is the distance from the centerline (vertical shift) to the peak. From the graph, the maximum value is 88 and the minimum value is 8-8, so the amplitude is: A=8(8)2=8A = \frac{8 - (-8)}{2} = 8

  2. Period and kk: The period is the distance between two consecutive peaks or troughs. Observing the graph, the peaks are at x=9x = -9 and x=3x = -3, meaning the period is: Period=3(9)=6\text{Period} = -3 - (-9) = 6 The relationship between the period and kk is given by: Period=2πk\text{Period} = \frac{2\pi}{k} So, solving for kk: 6=2πkk=2π6=π36 = \frac{2\pi}{k} \quad \Rightarrow \quad k = \frac{2\pi}{6} = \frac{\pi}{3}

  3. Phase Shift: The graph resembles a cosine wave, as it starts at a maximum at x=9x = -9. Since the general cosine function cos(kx)\cos(kx) has its peak at x=0x = 0, this graph is shifted to the left by 9 units. Therefore, the phase shift is 99.

  4. Vertical Shift CC: The graph oscillates symmetrically about y=0y = 0, so there is no vertical shift. Thus, C=0C = 0.

Final Equation:

The equation that matches the graph is: y=8cos(π3(x+9))y = 8 \cos\left( \frac{\pi}{3}(x + 9) \right)

Let me know if you need further details or explanations!


Here are 5 related questions you can explore:

  1. How do we calculate the amplitude for a trigonometric function?
  2. What is the general formula for the period of sine and cosine functions?
  3. How does the phase shift affect the graph of a trigonometric function?
  4. Can this graph be represented using a sine function instead of a cosine function?
  5. What role does the vertical shift CC play in modifying a trigonometric function?

Tip: The cosine function peaks at x=0x = 0, while the sine function starts from 0, which helps decide which form to use when modeling real-life periodic graphs.

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Math Problem Analysis

Mathematical Concepts

Trigonometry
Graph Analysis
Amplitude
Periodicity
Phase Shift
Vertical Shifts

Formulas

y = A sin(kx) + C
y = A cos(kx) + C
Amplitude = (max - min) / 2
Period T = 2π / k
Vertical Shift = C

Theorems

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Suitable Grade Level

Grades 9-12