Problem 59 - Algebra A Combined Function

Question

The average annual cinema admission price \(y\) (in dollars) from 2000 through 2008 is given by \(y=0.2 x+5.39 .\) In this equation, \(x\) represents the number of years after 2000. (Source: Motion Picture Association of America) a. Complete the table. $$ \begin{array}{|c|c|c|c|} \hline \boldsymbol{x} & 1 & 3 & 5 \\ \hline \boldsymbol{y} & & & \\ \hline \end{array} $$

Step-by-Step Solution

Answer

The values of \( y \) are 5.59, 5.99, and 6.39 for \( x = 1, 3, \) and \( 5 \), respectively.

Step 1: Identify the equation

The given equation for the average annual cinema admission price is \( y = 0.2x + 5.39 \). Here, \( x \) represents the number of years after the year 2000.

Step 2: Calculate \( y \) for \( x = 1 \)

Substitute \( x = 1 \) into the equation: \( y = 0.2 \times 1 + 5.39 \). Simplifying gives \( y = 0.2 + 5.39 = 5.59 \).

Step 3: Calculate \( y \) for \( x = 3 \)

Substitute \( x = 3 \) into the equation: \( y = 0.2 \times 3 + 5.39 \). Simplifying gives \( y = 0.6 + 5.39 = 5.99 \).

Step 4: Calculate \( y \) for \( x = 5 \)

Substitute \( x = 5 \) into the equation: \( y = 0.2 \times 5 + 5.39 \). Simplifying gives \( y = 1.0 + 5.39 = 6.39 \).

Step 5: Complete the table

Fill in the values of \( y \) from previous calculations into the table.\[\begin{array}{|c|c|c|c|}\hline \boldsymbol{x} & 1 & 3 & 5 \\hline \boldsymbol{y} & 5.59 & 5.99 & 6.39 \\hline\end{array}\]

Key Concepts

Function Evaluation

In mathematics, function evaluation is the process of determining the output of a function for a specific input. Let's look at the equation given in the exercise, which is a linear equation: \( y = 0.2x + 5.39 \). This function tells us the average annual cinema admission price \( y \) in dollars depending on the number of years \( x \) after 2000. Here, the function evaluation involves substituting a specific value of \( x \) into the equation to find \( y \).

For example, if \( x = 1 \), which represents the year 2001, we substitute 1 into the equation:

• Calculate: \( y = 0.2 \times 1 + 5.39 \)
• Simplify: \( y = 5.59 \)

This means that one year after 2000, the cinema admission price is \$5.59. By evaluating the function at different years \( x \), you can predict or calculate the price for those specific years.

Slope and Intercept

Linear equations like \( y = 0.2x + 5.39 \) are commonly expressed in the form \( y = mx + c \), where \( m \) is the slope and \( c \) is the y-intercept. These components are crucial to understand the trend and starting point of the function's graph.

• The **slope** \( m = 0.2 \) reflects the rate of change of the y-value with respect to x. It shows how much y increases for each additional year after 2000. Since the slope is positive, it indicates an increasing trend, meaning the cinema admission price is expected to rise.
• The **y-intercept** \( c = 5.39 \) represents the starting value of y when x is 0. This is the price at the year 2000. Therefore, \$5.39 is the baseline or initial cinema admission price at the beginning of the timeframe considered.

Understanding slope and intercept helps visualize how a linear function behaves and predict future values.

Table Completion

Completing a table involves calculating values of a function for specific inputs. In our exercise, we need to fill in the missing y-values for given x-values in the table format. This table allows for easy visualization and organization of the data points from the function.

• Start by substituting each value of x given in the table into the linear function \( y = 0.2x + 5.39 \).
• For \( x = 1 \), we found earlier that \( y = 5.59 \).
• For \( x = 3 \), subscribe into the equation: \( y = 0.2 \times 3 + 5.39 \), which simplifies to \( y = 5.99 \).
• For \( x = 5 \), substitute into the equation: \( y = 0.2 \times 5 + 5.39 \), which simplifies to \( y = 6.39 \).

Once calculated, you fill these results into the table under the corresponding x-values. This exercise helps reinforce the understanding of function evaluation and the impact of the linear relationship between the variables.