sara lovell 03/13/25 8:20 pm ? and cosine: vertical question part 7 of 7 completed: 9 of 13 | my score…

sara lovell 03/13/25 8:20 pm ? and cosine: vertical question part 7 of 7 completed: 9 of 13 | my score: 7.63/13 pts (58.67%) save the graph to the right is a function of the form y = a sin (bx - c)+d, b > 0. the five - quarter points of one cycle of the graph, from left to right, are given below. these five - quarter points on the graph correspond to the five - quarter points on the graph of y = sin x over the interval 0,2π. determine the equation of the specific function that is represented by the given graph based on the association of the labeled quarter points and the quarter points of the graph of y = sin x over the interval 0,2π. the quarter points are (π/4,0), (3π/4, - 5), (5π/4,0), (7π/4,5), and (9π/4,0). d = 0 (simplify your answer. type an exact answer, using π as needed. use integers or fractions for any numbers in the expression.) g. what is the function of the form y = a sin (bx - c)+d, where b > 0 and - π < c < π, that is represented by the given graph? (simplify your answer. type an exact answer, using π as needed. use integers or fractions for any numbers in the equation.)

sara lovell 03/13/25 8:20 pm ? and cosine: vertical question part 7 of 7 completed: 9 of 13 | my score: 7.63/13 pts (58.67%) save the graph to the right is a function of the form y = a sin (bx - c)+d, b > 0. the five - quarter points of one cycle of the graph, from left to right, are given below. these five - quarter points on the graph correspond to the five - quarter points on the graph of y = sin x over the interval 0,2π. determine the equation of the specific function that is represented by the given graph based on the association of the labeled quarter points and the quarter points of the graph of y = sin x over the interval 0,2π. the quarter points are (π/4,0), (3π/4, - 5), (5π/4,0), (7π/4,5), and (9π/4,0). d = 0 (simplify your answer. type an exact answer, using π as needed. use integers or fractions for any numbers in the expression.) g. what is the function of the form y = a sin (bx - c)+d, where b > 0 and - π < c < π, that is represented by the given graph? (simplify your answer. type an exact answer, using π as needed. use integers or fractions for any numbers in the equation.)

Answer

Explanation:

Step1: Find the amplitude A

The amplitude is half the distance between the maximum and minimum values. The maximum value is 5 and the minimum is - 5. So $A=\frac{5 - (-5)}{2}=\frac{10}{2}=5$.

Step2: Find the period P

The distance between two consecutive x - values of the same - type quarter - points (e.g., between $\frac{\pi}{4}$ and $\frac{9\pi}{4}$) is one period. $P=\frac{9\pi}{4}-\frac{\pi}{4}=2\pi$. Since the formula for the period of $y = A\sin(Bx - C)+D$ is $P=\frac{2\pi}{B}$, and $P = 2\pi$, then $B = 1$.

Step3: Find the phase - shift C

For the function $y=\sin x$, the first quarter - point is at $x = 0$. For the given function, the first quarter - point is at $x=\frac{\pi}{4}$. The phase - shift formula is $x=\frac{C}{B}$ for the first quarter - point of $y = A\sin(Bx - C)+D$. Since $B = 1$ and the first quarter - point of the given function is at $x=\frac{\pi}{4}$, then $C=\frac{\pi}{4}$.

Step4: Write the function

We know that $A = 5$, $B = 1$, $C=\frac{\pi}{4}$, and $D = 0$. Substituting these values into $y=A\sin(Bx - C)+D$, we get $y = 5\sin(x-\frac{\pi}{4})$.

Answer:

$y = 5\sin(x-\frac{\pi}{4})$