Which function has the given properties below?
The domain is the set of all real numbers.
One $x$-intercept is $\left(\frac{\pi}{2}, 0\right)$.
The maximum value is 3.
The $y$-intercept is $(0,-3)$.
A. $y=-3 \sin (x)$
B. $y=-3 \cos (x)$
C. $y=3 \sin (x)$
D. $y=3 \cos (x)$
Answer (2)
The function that meets all given properties is y = − 3 cos ( x ) , as it has the correct x -intercept, maximum value, and y -intercept. Therefore, the answer is option B.
;
y = − 3 cos ( x )
Explanation
Analyze the problem We are given four possible functions and four properties that the correct function must satisfy. We will check each function against each property to see which function satisfies all four properties.
Check each function against the properties
y = − 3 s in ( x ) :
Domain: All real numbers. This matches the given property.
x -intercept: When x = f r a c p i 2 , y = − 3 s in ( f r a c p i 2 ) = − 3 ( 1 ) = − 3 n e q 0 . This does not match the given property.
y = − 3 cos ( x ) :
Domain: All real numbers. This matches the given property.
x -intercept: When x = f r a c p i 2 , y = − 3 cos ( f r a c p i 2 ) = − 3 ( 0 ) = 0 . This matches the given property.
Maximum value: The maximum value of cos ( x ) is 1, so the maximum value of − 3 cos ( x ) is − 3 ( − 1 ) = 3 . This matches the given property.
y -intercept: When x = 0 , y = − 3 cos ( 0 ) = − 3 ( 1 ) = − 3 . This matches the given property.
y = 3 s in ( x ) :
Domain: All real numbers. This matches the given property.
x -intercept: When x = f r a c p i 2 , y = 3 s in ( f r a c p i 2 ) = 3 ( 1 ) = 3 n e q 0 . This does not match the given property.
y = 3 cos ( x ) :
Domain: All real numbers. This matches the given property.
x -intercept: When x = f r a c p i 2 , y = 3 cos ( f r a c p i 2 ) = 3 ( 0 ) = 0 . This matches the given property.
Maximum value: The maximum value of cos ( x ) is 1, so the maximum value of 3 cos ( x ) is 3 ( 1 ) = 3 . This matches the given property.
y -intercept: When x = 0 , y = 3 cos ( 0 ) = 3 ( 1 ) = 3 n e q − 3 . This does not match the given property.
Determine the correct function The function y = − 3 cos ( x ) satisfies all four properties.
Examples
Understanding trigonometric functions is crucial in many fields, such as physics and engineering. For example, when studying simple harmonic motion, like a pendulum swinging or a mass oscillating on a spring, the displacement can be modeled using sinusoidal functions. The properties of these functions, such as intercepts and maximum values, help engineers predict the behavior of these systems and design them effectively. By analyzing the function's domain, intercepts, and maximum values, we can accurately model and understand the motion of the object.
