Math Problem Statement

Reši nalogo.

Solution

The image you uploaded contains a graph of a trigonometric function, and the problem asks to determine the parameters AA, ω\omega, BB, and ϕ\phi for the general form of the function:

f(x)=Asin(ωx+ϕ)+Bf(x) = A \sin(\omega x + \phi) + B

Let's break down the graph to identify these parameters.

1. Amplitude AA:

  • The amplitude AA is the distance from the midline (average value) to the maximum or minimum value of the function.
  • Looking at the graph, the maximum value appears to be 33 and the minimum value is 1-1. Thus, the amplitude AA is: A=Max valueMin value2=3(1)2=42=2A = \frac{\text{Max value} - \text{Min value}}{2} = \frac{3 - (-1)}{2} = \frac{4}{2} = 2

2. Vertical Shift BB:

  • The vertical shift BB is the average value or the midline of the function.
  • This can be found by taking the average of the maximum and minimum values: B=Max value+Min value2=3+(1)2=22=1B = \frac{\text{Max value} + \text{Min value}}{2} = \frac{3 + (-1)}{2} = \frac{2}{2} = 1

3. Angular Frequency ω\omega:

  • The angular frequency ω\omega is related to the period TT of the function, where T=2πωT = \frac{2\pi}{\omega}.
  • From the graph, the period TT is the distance between two successive points where the function starts repeating. It appears that the period TT is 22, so: T=2ω=2πT=2π2=πT = 2 \quad \Rightarrow \quad \omega = \frac{2\pi}{T} = \frac{2\pi}{2} = \pi

4. Phase Shift ϕ\phi:

  • The phase shift ϕ\phi can be determined by observing where the function reaches a significant point (like the maximum or the zero crossing).
  • If we take the standard sine function, sin(0)=0\sin(0) = 0. In the graph, the function appears to cross the midline from below at x=0x = 0, so the phase shift ϕ\phi is 00.

Given this analysis, the function can be written as:

f(x)=2sin(πx)+1f(x) = 2 \sin(\pi x) + 1

Would you like further details or have any questions about this problem? Here are five related questions you could explore:

  1. How would the function change if the amplitude AA were doubled?
  2. What happens to the graph if the vertical shift BB is increased by 2?
  3. How does altering the period TT affect the angular frequency ω\omega?
  4. Can you describe the impact of a phase shift ϕ\phi on the sine function?
  5. How would the graph look if a cosine function were used instead of a sine function?

Tip: When analyzing trigonometric functions, always start by identifying the amplitude and period, as they directly affect the shape and frequency of the graph.

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Math Problem Analysis

Mathematical Concepts

Trigonometry
Graphical Analysis of Functions

Formulas

General form of trigonometric function f(x) = A sin(ωx + φ) + B
Period of trigonometric functions T = 2π/ω

Theorems

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Suitable Grade Level

Advanced High School