the graph of one complete period of a cosine curve is given. (a) find the amplitude, period, and horizontal…
the graph of one complete period of a cosine curve is given. (a) find the amplitude, period, and horizontal shift. (assume the absolute value of the horizontal shift is less than the amplitude) amplitude 8 period π horizontal shift 0 (b) write an equation that represents the curve in the form y = a cos(k(x - b)). y = need help? read it
Answer
Explanation:
Step1: Identify the values of a, k, and b
The general form of the cosine - function is $y = a\cos(k(x - b))$. We know from part (a) that the amplitude $a = 8$, the period $T=\pi$. The formula for the period of a cosine function is $T=\frac{2\pi}{k}$. Since $T = \pi$, we can solve for $k$. Also, the horizontal shift $b = 0$.
Step2: Solve for k
Given $T=\frac{2\pi}{k}$ and $T=\pi$, we substitute $T$ into the formula: $\pi=\frac{2\pi}{k}$. Cross - multiply to get $k\pi=2\pi$. Divide both sides by $\pi$ to find $k = 2$.
Step3: Write the equation
Substitute $a = 8$, $k = 2$, and $b = 0$ into the general form $y=a\cos(k(x - b))$. We get $y = 8\cos(2x)$.
Answer:
$y = 8\cos(2x)$