**Problem 4.3.41**Determine the simplest form of an equation for the graph shown. Choose $ b > 0 $, and include no phase shifts. (Midpoints and quarter-points are identified by dots).**An equation of the function shown is** $ y = \_\_\_ $---**Graph Description:**- The graph depicts a periodic function, likely a sine or cosine wave.- There are dots marking the midpoints and quarter-points on the curve.- The x-axis is labeled with critical points, including 0, $\frac{\pi}{2}$, and $\frac{3\pi}{2}$.- The y-axis is marked at increments of 2, both positively and negatively.**Function Characteristics:**- The function appears to oscillate between -4 and 4 on the y-axis.- Key points appear symmetrically distributed around the x-axis intercepts, suggesting the absence of a vertical shift.- The curve crosses the x-axis at x=0, indicating the function could be a sine or cosine with no horizontal phase shift.**Instructions for Solving:**To determine the equation:1. Identify the amplitude, which is half the distance between the maximum and minimum points.2. Determine the period from the graph, measuring the distance between two equivalent points (e.g., peaks or zero crossings).3. Calculate the value of $ b $ using the period ($ \text{Period} = \frac{2\pi}{b} $).4. Write the function in the form $ y = a \sin(bx) $ or $ y = a \cos(bx) $, depending on the starting point of the wave (zero crossing suggests sine; peak suggests cosine).Enter your answer in the answer box and then click "Check Answer." **Determine the simplest form of an equation for the graph shown. Choose b > 0, and include no phase shifts. (Midpoints and quarter-points are identified by dots).***An equation of the function shown is $ y = [ \text{Input Box} ] $***Graph Description:**The graph is a sinusoidal wave plotted on a coordinate grid. - **Axes:** The vertical axis is labeled $ y $ and the horizontal axis is labeled $ x $.- **Grid:** The grid is marked with intervals: the x-axis ranges approximately from $ -\pi $ to $ 3\pi $, while the y-axis ranges from -10 to 10.- **Function:** The sinusoidal curve oscillates between a maximum y-value of 10 and a minimum y-value of -10, suggesting an amplitude of 10.- **Characteristics:** The graph completes one full cycle over the interval from $ -\pi $ to $ \pi $, indicating the period of the function. There are points marked on the graph representing midpoints and quarter-points of the sinusoidal wave.**Instructions:**The task is to input the simplest trigonometric function that matches the characteristics of this sinusoidal graph. The form of the function will be $ y = a \sin(bx) $ or $ y = a \cos(bx) $ with specified values for amplitude and period, where $ b > 0 $. The function should not include any phase shifts.

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Determine the simplest form of an equation for the graph shown. Choose b>0, and include no phase shifts. (Midpoints and quarter-points are identified by dots). An equation of the function shown is y

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Transformation of Graphs

The word ‘transformation’ means modification. Transformation of the graph of a function is a process by which we modify or change the original graph and make a new graph.

Exponential Functions

The exponential function is a type of mathematical function which is used in real-world contexts. It helps to find out the exponential decay model or exponential growth model, in mathematical models. In this topic, we will understand descriptive rules, concepts, structures, graphs, interpreter series, work formulas, and examples of functions involving exponents.

Question

**Problem 4.3.41**

Determine the simplest form of an equation for the graph shown. Choose $ b > 0 $, and include no phase shifts. (Midpoints and quarter-points are identified by dots).

**An equation of the function shown is** $ y = \_\_\_ $

---

**Graph Description:**

- The graph depicts a periodic function, likely a sine or cosine wave.
- There are dots marking the midpoints and quarter-points on the curve.
- The x-axis is labeled with critical points, including 0, $\frac{\pi}{2}$, and $\frac{3\pi}{2}$.
- The y-axis is marked at increments of 2, both positively and negatively.

**Function Characteristics:**

- The function appears to oscillate between -4 and 4 on the y-axis.
- Key points appear symmetrically distributed around the x-axis intercepts, suggesting the absence of a vertical shift.
- The curve crosses the x-axis at x=0, indicating the function could be a sine or cosine with no horizontal phase shift.

**Instructions for Solving:**

To determine the equation:

1. Identify the amplitude, which is half the distance between the maximum and minimum points.
2. Determine the period from the graph, measuring the distance between two equivalent points (e.g., peaks or zero crossings).
3. Calculate the value of $ b $ using the period ($ \text{Period} = \frac{2\pi}{b} $).
4. Write the function in the form $ y = a \sin(bx) $ or $ y = a \cos(bx) $, depending on the starting point of the wave (zero crossing suggests sine; peak suggests cosine).

Enter your answer in the answer box and then click "Check Answer."

**Determine the simplest form of an equation for the graph shown. Choose b > 0, and include no phase shifts. (Midpoints and quarter-points are identified by dots).**

*An equation of the function shown is $ y = [ \text{Input Box} ] $*

**Graph Description:**
The graph is a sinusoidal wave plotted on a coordinate grid.

- **Axes:** The vertical axis is labeled $ y $ and the horizontal axis is labeled $ x $.
- **Grid:** The grid is marked with intervals: the x-axis ranges approximately from $ -\pi $ to $ 3\pi $, while the y-axis ranges from -10 to 10.
- **Function:** The sinusoidal curve oscillates between a maximum y-value of 10 and a minimum y-value of -10, suggesting an amplitude of 10.
- **Characteristics:** The graph completes one full cycle over the interval from $ -\pi $ to $ \pi $, indicating the period of the function. There are points marked on the graph representing midpoints and quarter-points of the sinusoidal wave.

**Instructions:**
The task is to input the simplest trigonometric function that matches the characteristics of this sinusoidal graph. The form of the function will be $ y = a \sin(bx) $ or $ y = a \cos(bx) $ with specified values for amplitude and period, where $ b > 0 $. The function should not include any phase shifts.

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