Problem 8 - Algebra 2

Question

For Exercises \(7-10\) , use the following information. The number of cable TV systems after 1985 can be modeled by the function \(C(t)=-43.2 t^{2}+1343 t+790,\) where \(t\) represents the number of years since \(1985 .\) Describe the turning points of the graph and its end behavior.

Step-by-Step Solution

Answer

The graph's turning point is a maximum at \( t \approx 15.55 \) years with \( C \approx 10426.1 \), and the end behavior is \( C(t) \to -\infty \) as \( t \to \pm \infty \).

Step 1: Identify the Type of Function

The given function \( C(t) = -43.2t^2 + 1343t + 790 \) is a quadratic function. This can be identified by the \( t^2 \) term, which indicates it's a parabola.

Step 2: Determine the Direction of the Parabola

Since the coefficient of \( t^2 \) is \(-43.2\) and is negative, the parabola opens downward.

Step 3: Find the Vertex

The vertex of a parabola \( ax^2+bx+c \) can be found using the formula \( t = -\frac{b}{2a} \). Here \( a = -43.2 \) and \( b = 1343 \). So \( t = -\frac{1343}{2(-43.2)} = \frac{1343}{86.4} \approx 15.55 \).

Step 4: Calculate the Value of C at the Vertex

Substitute \( t = 15.55 \) back into the function \( C(t) \) to calculate the maximum value. \[ C(15.55) = -43.2(15.55)^2 + 1343(15.55) + 790 \approx 10426.1 \].

Step 5: Determine the Turning Point

The turning point is the vertex of the quadratic function. It occurs at approximately \( t = 15.55 \) years and results in a maximum value of \( C \approx 10426.1 \). This indicates the number of cable TV systems peaked around the year \( 2000 \).

Step 6: Analyze the End Behavior

As \( t \to -\infty \), the term \( -43.2t^2 \) dominates and \( C(t) \to -\infty \). Similarly, as \( t \to \infty \), \( C(t) \to -\infty \) as well. This confirms the downwards-opening parabola ends going downward at both tails.

Key Concepts

Parabola

In mathematics, a parabola is a U-shaped curve that is defined by a quadratic function. Its general form is \( y = ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants. Parabolas are symmetric and can open either upwards or downwards.

• If the coefficient \( a \) is positive, the parabola opens upwards, resembling a smile.
• If \( a \) is negative, it opens downwards, like a frown.

In the given function \( C(t) = -43.2t^2 + 1343t + 790 \), the coefficient \( a = -43.2 \) is negative. Therefore, this parabola opens downward. This shape will help us visualize the function's behavior and make predictions about its highest and lowest points.

Vertex

The vertex of a parabola is an important point. It is either the maximum or minimum point on the graph, depending on whether the parabola opens upwards or downwards. It represents the peak or the trough of the curve.

• The vertex can be found using the formula \( t = -\frac{b}{2a} \) in the quadratic equation \( ax^2 + bx + c \).
• After finding \( t \), you substitute it back into the function to find the corresponding \( C \) value.

For the function \( C(t) = -43.2t^2 + 1343t + 790 \):

• The vertex \( t \) occurs at \( t = \frac{1343}{86.4} \approx 15.55 \).
• The \( C \) value at this vertex is approximately \( 10,426.1 \).

This vertex indicates that the maximum number of cable TV systems occurred around 15.55 years after 1985, which is approximately 2000.

End Behavior

End behavior describes how the values of a function behave as the input values become very large in the positive or negative direction. For quadratic functions, this behavior is determined by the leading coefficient \( a \).

• As \( t \to -\infty \), if \( a < 0 \), then \( C(t) \to -\infty \). This means the graph will fall indefinitely as you move left.
• As \( t \to \infty \), if \( a < 0 \), then \( C(t) \to -\infty \) again. This means the graph will also fall indefinitely as you move right.

In the given function:

The coefficient \( a = -43.2 \) is negative, which tells us that both ends of the parabola \( C(t) = -43.2t^2 + 1343t + 790 \) will point downwards. So, regardless of which direction you follow, the values of \( C(t) \) are decreasing.

Turning Point

The turning point of a parabola is synonymous with its vertex, which is the point where the graph changes direction. For the function \( C(t) = -43.2t^2 + 1343t + 790 \), the turning point is critical to understanding its overall behavior.

• The turning point occurs at the vertex \( t = 15.55 \), which coincides with the highest point of the parabola because it opens downward.
• This specific point is where the function value is maximum, which means it is the point where the number of cable TV systems peaks.

The turning point is crucial in many contexts, such as economics and physical sciences, as it helps identify points of maximum efficiency or peak demand. In this problem, knowing the turning point tells us exactly when cable TV systems reached their highest number after 1985.