Finding The Function Domain X-intercept Maximum Value And Y-intercept

Determining the correct function based on its properties is a fundamental skill in mathematics, particularly in trigonometry and calculus. In this article, we will explore how to identify a function given its domain, x-intercepts, maximum value, and y-intercept. We'll focus on the specific problem of finding a function that meets the following criteria:

• The domain is the set of all real numbers.
• One x-intercept is $(\frac{\pi}{2}, 0)$.
• The maximum value is 3.
• The y-intercept is $(0,-3)$.

We will examine three potential functions: A. $y=-3 \sin (x)$, B. $y=-3 \cos (x)$, and C. $y=3 \sin (x)$, and methodically analyze each to see if it fits the given properties. This process involves understanding the characteristics of sine and cosine functions, including their amplitudes, periods, intercepts, and how transformations affect their graphs. By the end of this exploration, you will have a clear understanding of how to match functions to their properties and a deeper appreciation for the interplay between algebraic representation and graphical behavior.

Understanding the Properties of Trigonometric Functions

To identify the correct function, we need a solid understanding of the properties of trigonometric functions, specifically sine and cosine. These functions are the backbone of periodic phenomena, and their characteristics dictate their behavior on the Cartesian plane. Let's delve into the key properties that will guide our analysis.

The sine function, denoted as $y = \sin(x)$, oscillates between -1 and 1. Its graph starts at the origin (0,0), increases to a maximum of 1 at $x = \frac{\pi}{2}$, returns to 0 at $x = \pi$, reaches a minimum of -1 at $x = \frac{3\pi}{2}$, and completes its cycle back at 0 at $x = 2\pi$. This cyclical behavior repeats indefinitely, making its domain the set of all real numbers. Crucially, the sine function has x-intercepts at integer multiples of $\pi$ (i.e., $0, \pi, 2\pi, -\pi$, etc.) and a y-intercept at (0,0).

In contrast, the cosine function, $y = \cos(x)$, also oscillates between -1 and 1, but it starts its cycle at a maximum of 1 when $x = 0$. It decreases to 0 at $x = \frac{\pi}{2}$, reaches a minimum of -1 at $x = \pi$, returns to 0 at $x = \frac{3\pi}{2}$, and completes its cycle at 1 at $x = 2\pi$. Like the sine function, its domain is all real numbers. However, the cosine function has x-intercepts at odd multiples of $\frac{\pi}{2}$ (i.e., $\frac{\pi}{2}, \frac{3\pi}{2}, -\frac{\pi}{2}$, etc.) and a y-intercept at (0,1).

The amplitude of a trigonometric function determines its maximum displacement from the x-axis. For the standard sine and cosine functions, the amplitude is 1. However, multiplying the function by a constant scales the amplitude. For instance, in $y = A\sin(x)$ or $y = A\cos(x)$, the amplitude is $|A|$. If $A$ is negative, the function is reflected across the x-axis.

The period of a trigonometric function is the length of one complete cycle. For both $y = \sin(x)$ and $y = \cos(x)$, the period is $2\pi$. Transformations such as horizontal stretches or compressions can alter the period. In the general forms $y = \sin(Bx)$ or $y = \cos(Bx)$, the period is given by $\frac{2\pi}{|B|}$.

Vertical shifts involve adding a constant to the function, moving the entire graph up or down. A function like $y = \sin(x) + C$ shifts the graph of $y = \sin(x)$ vertically by $C$ units. A positive $C$ shifts the graph upwards, while a negative $C$ shifts it downwards.

With these properties in mind, we can now methodically analyze the given functions to see which one matches the specified criteria.

Analyzing the Potential Functions

Now, let's methodically analyze each of the potential functions against the given properties:

• The domain is the set of all real numbers.
• One x-intercept is $(\frac{\pi}{2}, 0)$.
• The maximum value is 3.
• The y-intercept is $(0,-3)$.

Function A: $y = -3 \sin(x)$

This function is a sine function with an amplitude of $|-3| = 3$, reflected across the x-axis due to the negative coefficient. The domain of the sine function is all real numbers, so this condition is satisfied. The sine function typically has a y-intercept at (0,0), and this function also passes through the origin since $-3 \sin(0) = 0$. Thus, the y-intercept condition of (0, -3) is not met.

The x-intercepts of the standard sine function occur at integer multiples of $\pi$. In this case, $-3 \sin(x) = 0$ when $\sin(x) = 0$, which occurs at $x = n\pi$, where $n$ is an integer. Therefore, $\frac{\pi}{2}$ is not an x-intercept for this function. This also violates the given criteria.

Finally, the sine function oscillates between -1 and 1. Multiplying by -3 scales this range to [-3, 3], but the reflection across the x-axis means the function oscillates between 3 and -3. The maximum value of $y = -3\sin(x)$ is 3, which satisfies the third condition.

Conclusion for Function A: While it meets the domain and maximum value requirements, it fails to satisfy the x-intercept and y-intercept conditions. Therefore, Function A is not the correct answer.

Function B: $y = -3 \cos(x)$

This function is a cosine function with an amplitude of $|-3| = 3$, reflected across the x-axis. Like the sine function, the cosine function has a domain of all real numbers, so this condition is met. The standard cosine function has a y-intercept at (0,1). Multiplying by -3 transforms this into (0, -3), which satisfies the y-intercept condition.

The x-intercepts of the cosine function occur at odd multiples of $\frac{\pi}{2}$. In this case, $-3 \cos(x) = 0$ when $\cos(x) = 0$. This indeed occurs at $x = \frac{\pi}{2}$, satisfying the x-intercept condition.

The cosine function oscillates between -1 and 1. Multiplying by -3 scales this range to [-3, 3], and the reflection across the x-axis ensures the function oscillates between 3 and -3. The maximum value of $y = -3\cos(x)$ is 3, satisfying the maximum value condition.

Conclusion for Function B: This function satisfies all the given conditions: domain, x-intercept, maximum value, and y-intercept. Therefore, Function B is a strong candidate.

Function C: $y = 3 \sin(x)$

This function is a sine function with an amplitude of 3. The domain is all real numbers, which satisfies the first condition. However, the sine function typically has a y-intercept at (0,0), and this function also passes through the origin since $3 \sin(0) = 0$. This violates the y-intercept condition of (0, -3).

The x-intercepts of the standard sine function occur at integer multiples of $\pi$. Thus, $3 \sin(x) = 0$ when $\sin(x) = 0$, which occurs at $x = n\pi$, where $n$ is an integer. Therefore, $\frac{\pi}{2}$ is not an x-intercept for this function, violating another condition.

The sine function oscillates between -1 and 1. Multiplying by 3 scales this range to [-3, 3]. The maximum value of $y = 3\sin(x)$ is 3, satisfying the maximum value condition.

Conclusion for Function C: While it meets the domain and maximum value requirements, it fails to satisfy the x-intercept and y-intercept conditions. Therefore, Function C is not the correct answer.

Conclusion: Identifying the Correct Function

After a thorough analysis of each function, we have determined that:

• Function A ($y = -3 \sin(x)$) does not satisfy the x-intercept and y-intercept conditions.
• Function B ($y = -3 \cos(x)$) satisfies all the given conditions.
• Function C ($y = 3 \sin(x)$) does not satisfy the x-intercept and y-intercept conditions.

Therefore, the function that matches the given properties is Function B: $y = -3 \cos(x)$. This exercise demonstrates the importance of understanding the properties of trigonometric functions and how transformations affect their graphs. By systematically analyzing the given conditions, we can accurately identify the correct function.

This approach can be applied to various mathematical problems, highlighting the significance of connecting function properties with their algebraic representations. The ability to identify functions based on their characteristics is not only crucial for solving mathematical problems but also for understanding and modeling real-world phenomena that exhibit periodic behavior.