Problem 47 - Algebra 1: Concepts and Skills

Question

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

Step-by-Step Solution

Answer

The values of y when \(x=-2, -1, 0\) and \(1\) are -2.5, -1.5, -0.5 ,0.5 respectively.

Step 1: Evaluate the function when \(x=-2\)

First, replace \(x\) with \(-2\) in the function \(y=x-\frac{1}{2}\):\nUsing \(-2\), the function becomes \(y=-2-\frac{1}{2}=-2.5\).

Step 2: Evaluate the function when \(x=-1\)

Next, replace \(x\) with \(-1\) in the function:\nUsing \(-1\), the function becomes \(y=-1-\frac{1}{2}=-1.5\).

Step 3: Evaluate the function when \(x=0\)

Next, replace \(x\) with \(0\) in the function:\nUsing \(0\), the function becomes \(y=0-\frac{1}{2}=-0.5\).

Step 4: Evaluate the function when \(x=1\)

Finally, replace \(x\) with \(1\) in the function:\nUsing \(1\), the function becomes \(y=1-\frac{1}{2}=0.5\).

Step 5: Organize the results in a table

We can now organize these findings in a table. The table has two rows - 'x' and 'y'. We list down the x-values (-2,-1,0,1) along with their corresponding y-values (-2.5, -1.5, -0.5 ,0.5) respectively. The table thus looks like this:\n\[\begin{array}{|c|c|}\hlinex & y \\hline-2 & -2.5 \\hline-1 & -1.5 \\hline0 & -0.5 \\hline1 & 0.5 \\hline\end{array}\]

Key Concepts

Function Evaluation

Understanding how to evaluate a function is a fundamental skill in algebra. To evaluate a function means to find the value of the function's output (often represented as 'y') when given a specific input value (often represented as 'x'). This process is akin to substituting the value of x into the function and calculating the result.

In our example, the function given is \( y = x - \frac{1}{2} \). When we want to evaluate this function for \( x = -2 \), we replace x with -2, resulting in \( y = -2 - \frac{1}{2} \), which simplifies to \( y = -2.5 \). Similarly, evaluating the function for x-values of -1, 0, and 1 gives us respective y-values of -1.5, -0.5, and 0.5. Through this step-by-step substitution and simplification process, we determine the output values for various inputs, helping us understand the behavior of the function in relation to changes in x.

Linear Functions

Linear functions are a type of function where the graph is a straight line. These functions can always be written in the form \( y = mx + b \) where \( m \) is the slope, indicating the steepness of the line, and \( b \) is the y-intercept, indicating where the line crosses the y-axis.

The function \( y = x - \frac{1}{2} \) is a linear function because it fits this form; here, the slope (m) is 1 and the y-intercept (b) is \( -\frac{1}{2} \). This means for every unit increase in x, y increases by 1 unit, and when x is 0, y is \( -\frac{1}{2} \). Linear functions are useful for predicting and understanding relationships where change occurs at a constant rate.

Creating Value Tables

Creating a value table is a great way to organize the outputs of a function based on a range of inputs. This table consists of two columns, one for the independent variable (x) and one for the dependent variable (y). For our function \( y = x - \frac{1}{2} \), we create a value table by listing the chosen x-values and their corresponding calculated y-values.

This structure helps visualize the function's behavior across different values and is especially helpful for plotting these points on a graph. Value tables serve as a practical tool in analyzing the function systematically, for example, by observing patterns or trends in the outputs, which are essential for recognizing the characteristics of the function.