progress: question id: 141092 the movement of the progress bar may be uneven because questions can be worth…

progress: question id: 141092 the movement of the progress bar may be uneven because questions can be worth more or less (including zero) depending on your answer. the graph of a cosine function is drawn. one full cycle goes from x = 0 to x = 2 and the low point on that cycle is (1, - 5). which of the following functions could have this graph? o y = - 5 · cos 2πx o y = 5 · cos 2πx o y = 5 · cos πx o y = - 5 · cos πx

progress: question id: 141092 the movement of the progress bar may be uneven because questions can be worth more or less (including zero) depending on your answer. the graph of a cosine function is drawn. one full cycle goes from x = 0 to x = 2 and the low point on that cycle is (1, - 5). which of the following functions could have this graph? o y = - 5 · cos 2πx o y = 5 · cos 2πx o y = 5 · cos πx o y = - 5 · cos πx

Answer

Answer:

D. $y=- 5\cdot\cos(\pi x)$

Explanation:

Step1: Find the period formula

The general form of a cosine - function is $y = A\cos(Bx - C)+D$. The period $T$ of the cosine function $y = A\cos(Bx - C)+D$ is given by $T=\frac{2\pi}{|B|}$.

Step2: Calculate the period

We know that one full - cycle goes from $x = 0$ to $x = 2$, so the period $T = 2$. Using the period formula $T=\frac{2\pi}{|B|}$, and substituting $T = 2$, we get $2=\frac{2\pi}{|B|}$. Solving for $|B|$: [ \begin{align*} 2|B|&=2\pi\ |B|&=\pi \end{align*} ] So, $B=\pm\pi$.

Step3: Determine the amplitude

The amplitude of the cosine function $y = A\cos(Bx - C)+D$ is $|A|$. The low - point of the cosine function $y = A\cos(Bx - C)+D$ is $y = D - |A|$. We know that the low - point is $(1,-5)$. For a basic cosine function $y = A\cos(Bx)$, when $x$ is at the mid - point of the cycle (where the low - point occurs in a standard cosine function), $\cos(Bx)=- 1$. We know that the low - point of the function is $y=-5$. For the cosine function $y = A\cos(Bx)$, when $\cos(Bx)=-1$, $y=-A$. So, $A = 5$ or $A=-5$. Since the low - point of the cosine function occurs when $\cos(Bx)=-1$ and $y=-5$, we have $A = 5$ and the function is of the form $y=-5\cos(Bx)$ (because when $\cos(Bx)=1$, $y=-5\times1=-5$ at the low - point). Combining the results of the period and amplitude, the function is $y=-5\cos(\pi x)$.