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.

Solution

From the image, the problem is to determine a function of the form:

y=Asin(kx)+Cory=Acos(kx)+Cy = A \sin(kx) + C \quad \text{or} \quad y = A \cos(kx) + C

that matches the graph shown. Here's how we solve this:


Step-by-step Analysis:

  1. Determine the amplitude AA:

    • The amplitude is the distance from the centerline to the peak (or trough).
    • From the graph, the peak is at y=2y = 2 and the trough is at y=2y = -2. Hence, A=2A = 2.
  2. Determine the vertical shift CC:

    • The centerline is midway between the peak and the trough. This value is C=0C = 0, as the centerline is along y=0y = 0.
  3. Determine the period and kk:

    • The period of a sine or cosine function is given by: Period=2πk.\text{Period} = \frac{2\pi}{k}.
    • From the graph, one complete cycle occurs from x=0x = 0 to x=πx = \pi. Thus, the period is π\pi, and: k=2ππ=2.k = \frac{2\pi}{\pi} = 2.
  4. Choose sine or cosine:

    • At x=0x = 0, the graph starts at y=0y = 0 and rises, which matches the sine function. Therefore, we use: y=Asin(kx)+C.y = A \sin(kx) + C.

Final Answer:

The equation of the function is: y=2sin(2x).y = 2 \sin(2x).

Would you like a detailed explanation of any step? 😊


Related Questions:

  1. What is the difference between sine and cosine graphs?
  2. How do you calculate the period of a sine or cosine function?
  3. What is the significance of the vertical shift CC?
  4. How do you graph a sine function from its equation?
  5. How can we verify if a function matches a given graph?

Tip:

To easily identify kk, check how many cycles fit into 2π2\pi and use k=2πPeriodk = \frac{2\pi}{\text{Period}}.

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Math Problem Analysis

Mathematical Concepts

Trigonometry
Graphing Sine and Cosine Functions
Amplitude
Period
Vertical Shift

Formulas

y = A sin(kx) + C
y = A cos(kx) + C
Period = 2π / k

Theorems

Properties of Sine and Cosine Graphs

Suitable Grade Level

Grades 10-12