the graph of one complete period of a sine curve is given. (a) find the amplitude, period, and horizontal…
the graph of one complete period of a sine curve is given. (a) find the amplitude, period, and horizontal shift. (assume the absolute value of the horizontal shift is less than the period.) amplitude 7 period 8π horizontal shift 0 (b) write an equation that represents the curve in the form y = a sin(k(x - b)). y =
Answer
Explanation:
Step1: Recall the general - form formula
The general form of a sine function is $y = a\sin(k(x - b))$, where $|a|$ is the amplitude, $T=\frac{2\pi}{k}$ is the period, and $b$ is the horizontal shift. We know from part (a) that $a = 7$, $T = 8\pi$, and $b = 0$.
Step2: Find the value of $k$
Since $T=\frac{2\pi}{k}$ and $T = 8\pi$, we can solve for $k$ by setting $\frac{2\pi}{k}=8\pi$. Cross - multiply to get $2\pi=8\pi k$. Then, divide both sides by $8\pi$: $k=\frac{2\pi}{8\pi}=\frac{1}{4}$.
Step3: Write the equation
Substitute $a = 7$, $k=\frac{1}{4}$, and $b = 0$ into the general form $y = a\sin(k(x - b))$. We get $y = 7\sin(\frac{1}{4}x)$.
Answer:
$y = 7\sin(\frac{1}{4}x)$