Question

The number of cable TV systems after 1985 can be modeled by the function $C\left(t\right)=-43.2t^2+1343t+790$, where (t) represents the number of years since 1985. Describe the turning points of the graph and its end behaviour.

Short answer

The turning point will be in between $t=15\text{and}t=16$.

End behavior of the graph is written as:As $t\to -\infty ,C\left(t\right)\to -\infty \text{and as}t\to \infty ,C\left(t\right)\to -\infty$.

Step-by-step solution

Step 1. Write down the given information.

The number of cable TV systems are modeled by the function $C\left(t\right)=-43.2t^2+1343t+790$, where (t) represents the number of years since 1985.

Step 2. Sketch the graph of given equation.

The graph of the equation $C\left(t\right)=-43.2t^2+1343t+790$ from the year 1985 to 2005 is shown below, where (t) represents the number of years since 1985.

Step 3. Analyzing the turning points and end behavior of the graph.

From the graph of the equation, it can be interpreted that the number of cable TV in between $t=15\text{and}t=16$ are maximum. So relative maxima is in between $t=15\text{and}t=16$.Hence the turning point will be in between $t=15\text{and}t=16$.Since the graph continues after $t=20$, so there will be no relative minima.

End behavior of the graph is written as: As $t\to -\infty ,C\left(t\right)\to -\infty \text{and as}t\to \infty ,C\left(t\right)\to -\infty$.

Step 4. Conclusion.

The turning point will be in between $t=15\text{and}t=16$.

End behavior of the graph is written as: As $t\to -\infty ,C\left(t\right)\to -\infty \text{and as}t\to \infty ,C\left(t\right)\to -\infty$.