Identify Function With Y-Intercept At -1 And Amplitude Of 2
Finding the right trigonometric function that fits specific criteria like a y-intercept of -1 and an amplitude of 2 requires a solid understanding of trigonometric functions and their properties. This article will methodically analyze the given options to determine which function meets these conditions. Understanding amplitude, y-intercepts, and the general forms of sine and cosine functions is key to solving this type of problem. Let's explore how each component of a trigonometric function affects its graph and identify the correct function step by step.
Understanding Amplitude and Y-Intercept
To identify the function with a y-intercept of -1 and an amplitude of 2, it's crucial to understand what these terms mean in the context of trigonometric functions. The amplitude of a trigonometric function, such as sine or cosine, determines the height of the wave from its center line (or midline). It is the absolute value of the coefficient multiplied by the trigonometric function. For example, in the function f(x) = A sin(x) or f(x) = A cos(x), |A| represents the amplitude. An amplitude of 2 means the function's graph will oscillate 2 units above and 2 units below its midline.
The y-intercept is the point where the function's graph intersects the y-axis. This occurs when x = 0. To find the y-intercept, you substitute x = 0 into the function and solve for y. For instance, if f(x) = -2cos(x) - 1, the y-intercept is f(0) = -2cos(0) - 1. Since cos(0) = 1, the y-intercept is -2(1) - 1 = -3. This calculation highlights the importance of understanding both the amplitude and y-intercept when analyzing trigonometric functions. To correctly identify the function, we must check each option to see which one satisfies both conditions: a y-intercept of -1 and an amplitude of 2. This involves careful evaluation of each function at x = 0 and a clear understanding of how the coefficients affect the amplitude and vertical shift of the graph. By breaking down each component and its impact, we can methodically determine the correct function.
Analyzing the Options
When trying to identify the function with a specific y-intercept and amplitude, each option needs to be carefully examined. This section breaks down each given function, calculating its y-intercept and amplitude to determine if it fits the criteria. The process involves substituting x = 0 to find the y-intercept and identifying the coefficient of the trigonometric function to determine the amplitude. For each function, we will go through these steps in detail.
1. f(x) = -sin(x) - 1
To find the y-intercept of f(x) = -sin(x) - 1, substitute x = 0:
f(0) = -sin(0) - 1
Since sin(0) = 0:
f(0) = -0 - 1 = -1
The y-intercept for this function is -1, which meets one of our criteria. Now, let's determine the amplitude. The coefficient of the sin(x) term is -1. The amplitude is the absolute value of this coefficient:
Amplitude = |-1| = 1
The amplitude of this function is 1, which does not match the required amplitude of 2. Therefore, this function does not fully meet the specified criteria.
2. f(x) = -2sin(x) - 1
To find the y-intercept of f(x) = -2sin(x) - 1, substitute x = 0:
f(0) = -2sin(0) - 1
Since sin(0) = 0:
f(0) = -2(0) - 1 = -1
The y-intercept for this function is -1, which meets our first criterion. Now, let's find the amplitude. The coefficient of the sin(x) term is -2. The amplitude is the absolute value of this coefficient:
Amplitude = |-2| = 2
The amplitude of this function is 2, which matches the required amplitude. So, this function satisfies both the y-intercept and amplitude conditions.
3. f(x) = -cos(x)
To find the y-intercept of f(x) = -cos(x), substitute x = 0:
f(0) = -cos(0)
Since cos(0) = 1:
f(0) = -1
The y-intercept for this function is -1, which meets one of the requirements. Now, let's find the amplitude. The coefficient of the cos(x) term is -1. The amplitude is the absolute value of this coefficient:
Amplitude = |-1| = 1
The amplitude of this function is 1, which does not match the required amplitude of 2. Therefore, this function does not fully meet the specified criteria.
4. f(x) = -2cos(x) - 1
To find the y-intercept of f(x) = -2cos(x) - 1, substitute x = 0:
f(0) = -2cos(0) - 1
Since cos(0) = 1:
f(0) = -2(1) - 1 = -3
The y-intercept for this function is -3, which does not match the required y-intercept of -1. Thus, this function does not meet our criteria. Although we can still calculate the amplitude for completeness, it's not necessary since the y-intercept already disqualifies it. The coefficient of the cos(x) term is -2. The amplitude is the absolute value of this coefficient:
Amplitude = |-2| = 2
The amplitude of this function is 2, which matches the required amplitude. However, because the y-intercept does not match, this function is not the correct answer.
Determining the Correct Function
After analyzing each function and calculating both their y-intercepts and amplitudes, it’s clear that one function stands out as the correct answer. This section will summarize the findings and definitively identify the function that meets the specified criteria: a y-intercept of -1 and an amplitude of 2. By revisiting the calculations and results from the previous section, we can confirm which function satisfies both conditions.
We evaluated four functions:
- f(x) = -sin(x) - 1
- f(x) = -2sin(x) - 1
- f(x) = -cos(x)
- f(x) = -2cos(x) - 1
Summary of Findings
- Function 1: f(x) = -sin(x) - 1 had a y-intercept of -1 and an amplitude of 1. While the y-intercept was correct, the amplitude did not match the required value of 2.
- Function 2: f(x) = -2sin(x) - 1 had a y-intercept of -1 and an amplitude of 2. This function met both criteria.
- Function 3: f(x) = -cos(x) had a y-intercept of -1 and an amplitude of 1. Again, the y-intercept was correct, but the amplitude did not match the required value of 2.
- Function 4: f(x) = -2cos(x) - 1 had a y-intercept of -3 and an amplitude of 2. Although the amplitude was correct, the y-intercept did not match the required value of -1.
Conclusion
Based on the analysis, the function that has a y-intercept at -1 and an amplitude of 2 is:
f(x) = -2sin(x) - 1
This function is the only one that satisfies both conditions, making it the correct answer. The detailed evaluation of each function allowed us to systematically eliminate the incorrect options and identify the function that perfectly matches the given criteria. Understanding how amplitude and y-intercept are determined for trigonometric functions is essential for solving problems of this nature.
Conclusion
In conclusion, the process of identifying the function with a specific y-intercept and amplitude requires a clear understanding of trigonometric functions and their properties. Through detailed analysis, we evaluated four options and determined that the function f(x) = -2sin(x) - 1 is the only one that meets both criteria: a y-intercept of -1 and an amplitude of 2. This exercise highlights the importance of knowing how coefficients affect the graph of trigonometric functions, particularly in terms of vertical shifts and amplitude changes. Understanding these concepts is crucial for solving similar problems and for a deeper comprehension of trigonometric functions in general. By breaking down each component of the functions and systematically checking against the given conditions, we successfully identified the function that fits the criteria.
