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)
Check if the domain of each function is the set of all real numbers.
Substitute x = 2 π into each function to check if y = 0 .
Determine the maximum value of each function and check if it is 3.
Substitute x = 0 into each function to check if y = − 3 .
The function that satisfies all four conditions is y = − 3 cos ( x ) .
Explanation
Understanding the Problem We are given four possible functions and four properties that the correct function must satisfy. Our goal is to determine which of the given functions satisfies all the given properties.
Listing the Properties and Candidate Functions The properties are:
The domain is the set of all real numbers.
One x -intercept is ( 2 π , 0 ) .
The maximum value is 3.
The y -intercept is ( 0 , − 3 ) .
The candidate functions are: y = − 3 sin ( x ) y = − 3 cos ( x ) y = 3 sin ( x ) y = 3 cos ( x )
Checking the Domain All given functions have a domain of all real numbers, so we proceed to the next property.
Checking the x-intercept We need to check which function has an x -intercept at ( 2 π , 0 ) . This means that when x = 2 π , y must be 0.
For y = − 3 sin ( x ) : y = − 3 sin ( 2 π ) = − 3 ( 1 ) = − 3 = 0 .
For y = − 3 cos ( x ) : y = − 3 cos ( 2 π ) = − 3 ( 0 ) = 0 . This function satisfies the x -intercept property. For y = 3 sin ( x ) : y = 3 sin ( 2 π ) = 3 ( 1 ) = 3 = 0 .
For y = 3 cos ( x ) : y = 3 cos ( 2 π ) = 3 ( 0 ) = 0 . This function satisfies the x -intercept property.
Checking the Maximum Value Now we check which of the remaining functions, y = − 3 cos ( x ) and y = 3 cos ( x ) , has a maximum value of 3.
The maximum value of y = − 3 cos ( x ) is 3, which occurs when cos ( x ) = − 1 .
The maximum value of y = 3 cos ( x ) is 3, which occurs when cos ( x ) = 1 . Both functions satisfy this property.
Checking the y-intercept Finally, we check which of the remaining functions, y = − 3 cos ( x ) and y = 3 cos ( x ) , has a y -intercept of ( 0 , − 3 ) . This means that when x = 0 , y must be − 3 .
For y = − 3 cos ( x ) : y = − 3 cos ( 0 ) = − 3 ( 1 ) = − 3 . This function satisfies the y -intercept property. For y = 3 cos ( x ) : y = 3 cos ( 0 ) = 3 ( 1 ) = 3 = − 3 . This function does not satisfy the y -intercept property.
Conclusion Therefore, the function that satisfies all the given properties is y = − 3 cos ( x ) .
Examples
Understanding trigonometric functions is crucial in many real-world applications, such as modeling periodic phenomena like sound waves, light waves, and alternating current (AC) circuits. For example, the function y = − 3 cos ( x ) could represent the voltage in an AC circuit, where the voltage oscillates between -3 and 3 volts. By analyzing the properties of this function, such as its intercepts and maximum value, engineers can design and optimize electrical systems for various applications.
The function that satisfies all given properties is y = − 3 cos ( x ) , as it has the specified intercepts and maximum value. It has a domain of all real numbers, crosses the x-axis at ( 2 θ , 0 ) , achieves a maximum value of 3, and has a y -intercept at (0, -3).
;
