Which function will have a $y$-intercept at -1 and an amplitude of $2 ?$
A. $f(x)=-\sin (x)-1$
B. $f(x)=-2 \sin (x)-1$
C. $f(x)=-\cos (x)$
D. $f(x)=-2 \cos (x)-1$
Answer (2)
The y -intercept is found by evaluating the function at x = 0 .
The amplitude of a sinusoidal function is the absolute value of the coefficient of the sine or cosine term.
Evaluate the y -intercept and amplitude for each function.
The function f ( x ) = − 2 sin ( x ) − 1 has a y -intercept of -1 and an amplitude of 2, thus the answer is f ( x ) = − 2 sin ( x ) − 1 .
Explanation
Understanding the Problem We are given four functions and we need to determine which one has a y -intercept of -1 and an amplitude of 2. The y -intercept is the value of the function when x = 0 , and the amplitude is the absolute value of the coefficient of the trigonometric function.
Analyzing Each Function Let's analyze each function:
f ( x ) = − sin ( x ) − 1 :
y -intercept: f ( 0 ) = − sin ( 0 ) − 1 = − 0 − 1 = − 1
Amplitude: ∣ − 1∣ = 1
f ( x ) = − 2 sin ( x ) − 1 :
y -intercept: f ( 0 ) = − 2 sin ( 0 ) − 1 = − 2 ( 0 ) − 1 = − 1
Amplitude: ∣ − 2∣ = 2
f ( x ) = − cos ( x ) :
y -intercept: f ( 0 ) = − cos ( 0 ) = − 1
Amplitude: ∣ − 1∣ = 1
f ( x ) = − 2 cos ( x ) − 1 :
y -intercept: f ( 0 ) = − 2 cos ( 0 ) − 1 = − 2 ( 1 ) − 1 = − 3
Amplitude: ∣ − 2∣ = 2
Identifying the Correct Function From the analysis above, we can see that the function f ( x ) = − 2 sin ( x ) − 1 has a y -intercept of -1 and an amplitude of 2.
Final Answer Therefore, the function that satisfies the given conditions is f ( x ) = − 2 sin ( x ) − 1 .
Examples
Understanding trigonometric functions like sine and cosine is crucial in many real-world applications. For instance, in electrical engineering, alternating current (AC) voltage and current are often modeled using sinusoidal functions. The amplitude represents the peak voltage or current, while the y-intercept can represent an initial offset. Similarly, in acoustics, sound waves can be modeled using trigonometric functions, where the amplitude corresponds to the loudness of the sound. Analyzing these functions helps engineers design circuits and audio equipment effectively.
The function f ( x ) = − 2 sin ( x ) − 1 has a y -intercept of -1 and an amplitude of 2, making it the correct answer. The analysis of each function confirmed that only Option B meets both criteria.
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