Problem 45 - Algebra 1: Concepts and Skills

Question

Evaluate the function when \(x=-2,-1,0\) and \(1 .\) Organize your results in a table. $$ y=-8.5-x $$

Step-by-Step Solution

Answer

The values of the function evaluated at \(x = -2, -1, 0, 1\) are \(y = -6.5, -7.5, -8.5, -9.5\), respectively.

Step 1: Function Evaluation for x=-2

Substitute \(x = -2\) into the equation. This will give \(y = -8.5 - (-2) = -8.5 + 2 = -6.5\)

Step 2: Function Evaluation for x=-1

Substitute \(x = -1\) into the equation. This will give \(y = -8.5 - (-1) = -8.5 + 1 = -7.5\)

Step 3: Function Evaluation for x=0

Substitute \(x = 0\) into the equation. This will give \(y = -8.5 - 0 = -8.5\)

Step 4: Function Evaluation for x=1

Substitute \(x = 1\) into the equation. This will give \(y = -8.5 - 1 = -9.5\)

Step 5: Organize in a Table

List all the values orderly in a table. \n\n\[\begin{tabular}{|c c|} \\ \hlinex & y \\ [0.5ex] \hline\hline-2 & -6.5 \\ -1 & -7.5 \\ 0 & -8.5 \\ 1 & -9.5 \\ [1ex] \hline\end{tabular}\]

Key Concepts

Linear Functions

Linear functions are some of the most basic and widely used mathematical functions. They can be easily recognized by their straight-line graph and are represented by the standard form equation, \( y = mx + b \), where \( m \) stands for the slope of the line and \( b \) represents the y-intercept, the point at which the line crosses the y-axis.

In essence, the slope \( m \) determines how steep the line is, and the y-intercept \( b \) indicates the starting point of the line on the graph. For every unit increase in \( x \), the value of \( y \) increases by \( m \) units. It's the simplicity and predictability of linear functions that make them so valuable in algebra to model relationships with a constant rate of change.

Function Notation

Function notation is a streamlined way of writing out equations that describe a function, with \( f(x) \) being the most common notation used to represent a function of \( x \). This notation is powerful; it tells us that for each input \( x \), there is an output \( y \) that corresponds to it.

For instance, the function in the exercise is \( y = -8.5 - x \), but it could also be written as \( f(x) = -8.5 - x \). When evaluating this function for a specific value, such as \( x = 1 \), we would plug in that value to get \( f(1) = -8.5 - 1 = -9.5 \). Function notation isn't just symbolic; it's a concise way of showing the operation to be performed on any input value to obtain the corresponding output, which is a critical aspect of understanding and analyzing functions.

Creating Value Tables

Creating value tables is a practical approach to organize and visualize how a function behaves for different inputs. It's a methodical process where you list chosen inputs (usually \( x \) values), calculate the corresponding outputs (\( y \) values), and then tabulate them in an easy-to-read format.

To construct a value table for a function, you'll:

• Decide upon a range of \( x \) values to evaluate, often centered around special points like where \( x = 0 \) or where the function may intersect with other functions.
• Plug each \( x \) value into the function to find the resultant \( y \) value.
• Record each \( x \) and its corresponding \( y \) in a two-column table, with the first column for \( x \) and the second for \( y \) values.

The table you generate becomes a pivotal tool in understanding the function's behavior without needing to graph it, although it can also aid in plotting points if a graph is required.