the graph of one complete period of a cosine curve is given. find the amplitude and period. amplitude 9…
the graph of one complete period of a cosine curve is given. find the amplitude and period. amplitude 9 thats it! period π fantastic! write an equation that represents the curve in the form y = a cos(k(x - b)). y = 1.5 cos(3/π π) × check your amplitude, period (or frequency), and phase shift. need help? read it
Answer
Explanation:
Step1: Determine the amplitude
The amplitude $a$ is the maximum distance from the maximum (or minimum) value of the function to the mid - line. Given the maximum value is 9 and the minimum is - 9, the mid - line is $y = 0$, so $a=9$.
Step2: Determine the period
The period $T$ is the length of one complete cycle. Here, the cycle goes from $x=\frac{\pi}{4}$ to $x = \frac{3\pi}{4}$, and $T=\frac{3\pi}{4}-\frac{\pi}{4}=\pi$. The formula for the period of $y = a\cos(k(x - b))$ is $T=\frac{2\pi}{k}$. Since $T = \pi$, we have $\pi=\frac{2\pi}{k}$, solving for $k$ gives $k = 2$.
Step3: Determine the phase - shift
The cosine function $y=\cos x$ has a maximum at $x = 0$. Our function has a maximum at $x = 0$. So the phase - shift $b = 0$.
Answer:
$y=9\cos(2x)$