Math Problem Statement

Write an equation of the form y = a sin(bx) or y = a cos(bx) to describe the graph shown in the image.

Solution

From the graph provided, we need to determine the equation of the form y=asin(bx)y = a \sin(bx) or y=acos(bx)y = a \cos(bx). Here's the step-by-step solution:


Step 1: Identify the amplitude

The amplitude aa is the maximum value of the function from the centerline (y = 0).

  • From the graph, the maximum value is 4 and the minimum value is -4.
  • Thus, the amplitude a=4a = 4.

Step 2: Determine the period

The period TT is the length of one complete wave. The formula for the period is: T=2πbT = \frac{2\pi}{b} From the graph:

  • The wave completes one full cycle between x=0x = 0 and x=2πx = 2 \pi. Thus, the period T=2πT = 2\pi.
  • Substituting T=2πT = 2\pi into the formula: 2π=2πb    b=12\pi = \frac{2\pi}{b} \implies b = 1

Step 3: Determine if it is sine or cosine

  • At x=0x = 0, the graph starts at 0 and rises upward, which matches the behavior of the sine function.
  • Therefore, the function is of the form y=asin(bx)y = a \sin(bx).

Final Answer:

The equation is: y=4sin(x)y = 4 \sin(x)


Would you like a deeper explanation or any verification steps?

Here are 5 related questions for practice:

  1. How would the equation change if the amplitude were halved?
  2. What happens to the graph if bb is doubled?
  3. How do you convert a sine equation to a cosine equation?
  4. What effect does a negative amplitude (a<0a < 0) have on the graph?
  5. How would the equation change if the graph started at its maximum point instead of 0?

Tip: When identifying the type of trigonometric function, always pay attention to where the graph starts at x=0x = 0. Sine starts at 0, while cosine starts at its maximum or minimum.

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Math Problem Analysis

Mathematical Concepts

Trigonometric Functions
Sine and Cosine Waves
Amplitude
Period

Formulas

y = a sin(bx)
y = a cos(bx)
Period T = 2π/b

Theorems

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Suitable Grade Level

Grades 9-12