2. graph and label both of the curves below. then, state their intersection points (in other words, solve…

2. graph and label both of the curves below. then, state their intersection points (in other words, solve the system of equations shown below).\n\n$y = 4\\cos(x)+1$\n\n$y=-\\cos(x)-4$\n\nintersection points:

2. graph and label both of the curves below. then, state their intersection points (in other words, solve the system of equations shown below).\n\n$y = 4\\cos(x)+1$\n\n$y=-\\cos(x)-4$\n\nintersection points:

Answer

Explanation:

Step1: Set the two equations equal

Since at intersection points $y$ - values are equal, we set $4\cos(x)+1 =-\cos(x)-4$.

Step2: Solve for $\cos(x)$

Add $\cos(x)$ to both sides: $4\cos(x)+\cos(x)+1=-\cos(x)+\cos(x)-4$, which simplifies to $5\cos(x)+1=-4$. Then subtract 1 from both sides: $5\cos(x)=-5$, so $\cos(x)=- 1$.

Step3: Find the values of $x$

We know that $\cos(x)=-1$ when $x=(2n + 1)\pi$, where $n\in\mathbb{Z}$. For the domain shown in the graph (usually $[-2\pi,2\pi]$), when $n = - 1,x=-\pi$; when $n = 0,x=\pi$.

Step4: Find the corresponding $y$ - values

Substitute $x = \pi$ into $y = 4\cos(x)+1$. Since $\cos(\pi)=-1$, then $y=4\times(-1)+1=-3$. The intersection points are of the form $(x,y)$.

Answer:

$(-\pi,-3),(\pi,-3)$