Problem 7 - 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 \(1985 .\) Graph this equation for the years 1985 to 2005 .

Step-by-Step Solution

Answer

Graph the quadratic function \(C(t) = -43.2t^2 + 1343t + 790\) from \(t = 0\) to \(t = 20\) and plot key calculated points.

Step 1: Identify Variables and Range

First, recognize that the function \( C(t) = -43.2t^2 + 1343t + 790 \) models the number of cable TV systems, with \( t \) representing the years since 1985. We need to find \( C(t) \) from 1985 to 2005, which corresponds to \( t = 0 \) to \( t = 20 \).

Step 2: Calculate Key Points

Calculate values of the function at key points for plotting. For instance, compute \( C(0) = 790 \), \( C(10) \), and \( C(20) \). These calculations help us understand the trend of the graph more effectively.

Step 3: Compute Intermediate Values

To plot a more accurate graph, calculate additional values throughout the domain, such as \( C(5), C(15), \) etc. This will provide points to ensure the curve is correctly shaped. You may use a calculator or software to find these values.

Step 4: Plotting the Graph

On a coordinate plane, plot the calculated points as \((t, C(t))\) where \(t\) is the horizontal axis and \(C(t)\) the vertical axis. Connect these points smoothly to reflect the quadratic nature of the equation, ensuring the curve extends from \(t = 0\) to \(t = 20\).

Step 5: Analyze the Graph

Observe the plotted graph to examine the number of cable TV systems over the given years. Note where the maximum or minimum points occur, any symmetry, and how the graph changes over time.

Key Concepts

Graphing Quadratic Equations

Graphing quadratic equations is a crucial skill in understanding their behavior and characteristics. A quadratic equation is typically in the form of \( ax^2 + bx + c \). The solution provided uses the specific function \( C(t) = -43.2t^2 + 1343t + 790 \), where \( t \) represents the number of years since 1985.
This equation models the number of cable TV systems over time. To graph a quadratic equation accurately:

• Determine the range of \( t \). For this problem, \( t \) varies from \( 0 \) to \( 20 \), representing the years from 1985 to 2005.
• Calculate key points by evaluating \( C(t) \) at different values like \( C(0), C(10), C(20) \). These points help you define the shape of the parabola on the graph.
• Graph these points on a coordinate plane with \( t \) on the horizontal axis and \( C(t) \) on the vertical axis.
• Connect the points smoothly, revealing the upside-down parabola caused by the negative \( a \)-value (\(-43.2\) in this case).

This visual representation will allow you to see changes over time, calculate estimates, and note parabolic symmetry.

Modeling with Quadratic Functions

Using quadratic functions to model real-world phenomena, like the number of cable TV systems, is common in mathematics due to their ability to represent various scenarios involving acceleration or growth followed by declination.
Quadratic functions provide valuable insights in situations where patterns show a rapid increase to a peak before a decrease, which is indicated by the parabola in the equation. The model \( C(t) = -43.2t^2 + 1343t + 790 \) forecasts changes over time in a specific context.
To successfully model with quadratic functions:

• Identify the variable, often representing time or another measurable change, giving context to \( t \).
• Recognize the coefficients' roles: the negative \( a \)-value points to a downward opening parabola, reflects eventual decrease.
• Calculate how \( b \) (1343) and \( c \) (790) influence the model, affecting progression and the starting point.

Such models help predict trends and evaluate decisions in planning and analysis, especially when examining historical data like technological adoption.

Function Analysis

Function analysis involves examining the properties and behaviors of functions. In the context of the equation \( C(t) = -43.2t^2 + 1343t + 790 \), understanding its components helps paint a broader picture about trends in cable TV systems.
Analyzing a quadratic function requires:

• Identifying critical points: These include the vertex, intercepts, and considering symmetry. Here, the vertex would indicate the peak year for the number of cable systems.
• Calculating the vertex using the formula \( t = \frac{-b}{2a} \), giving the time of peak systems, which translates into real-world scenarios.
• Understanding the direction of the parabola (downward in this instance), informs us about the declining trend post-peak as \( t \) increases towards the boundary of the domain.
• Evaluating how changes in \( t \) affect the number of systems, thus predicting potential increases or decreases and their implications.

This analysis component allows for a deeper insight into the applied equation, offering a thorough understanding of past trends and projecting future occurrences.