Problem 46 - Algebra 1: Concepts and Skills

Question

Evaluate the function when \(x=-2,-1,0\) and \(1 .\) Organize your results in a table. $$ y=-x-12.1 $$

Step-by-Step Solution

Answer

The summarized table of results is: For \(x = -2, -1, 0, 1\) the corresponding \(y\) values are -10.1, -11.1, -12.1 and -13.1 respectively.

Step 1: Understanding the Function

The function given is a linear function \(y = -x - 12.1\) where \(x\) is the input and \(y\) is the output.

Step 2: Evaluate for \(x = -2\)

Substitute \(x = -2\) into the function: \(y = -(-2) - 12.1 = 2 - 12.1 = -10.1\)

Step 3: Evaluate for \(x = -1\)

Substitute \(x = -1\) into the function: \(y = -(-1) - 12.1 = 1 - 12.1 = -11.1\)

Step 4: Evaluate for \(x = 0\)

Substitute \(x = 0\) into the function: \(y = -(0) - 12.1 = -12.1\)

Step 5: Evaluate for \(x = 1\)

Substitute \(x = 1\) into the function: \(y = -(1) - 12.1 = -1 - 12.1 = -13.1\)

Step 6: Summary Table of Results

Organize the \(x\) and \(y\) values in a table: \n \n | \(x\) | \(y\) |\n |--------------|--------------|\n |-2 | -10.1 |\n |-1 | -11.1 |\n | 0 | -12.1 |\n | 1 | -13.1 |

Key Concepts

linear functions

Linear functions are a fundamental concept in algebra, representing relationships that graph as straight lines. They have the general form: \[ y = mx + b \]Where:

• \( y \) is the dependent variable (output)
• \( m \) is the slope of the line
• \( x \) is the independent variable (input)
• \( b \) is the y-intercept

In this particular exercise, the function given is \( y = -x - 12.1 \), which is in a linear form where the slope \( m = -1 \) and the y-intercept \( b = -12.1 \). This indicates that for every unit increase in \( x \), \( y \) decreases by 1 unit, and the line crosses the y-axis at -12.1.
Linear functions are known for their constant rate of change, which in practical terms means whatever the increase or decrease in \( x \), the change in \( y \) will be consistent.

function evaluation

Function evaluation is the process of finding the output of a function for a given input. It involves substituting a number into the function and performing algebraic operations to find the result.
For instance, in our function \( y = -x - 12.1 \), to evaluate at \( x = -2 \), we substitute \( -2 \) into \( x \) in the function:
\( y = -(-2) - 12.1 = 2 - 12.1 = -10.1 \).
This is repeated for each given value of \( x \). It's a simple but powerful tool in algebra that allows us to understand how changing variables affect outcomes.

• Identify the value of \( x \).
• Substitute this value into the equation.
• Solve by following the order of operations: Parentheses first, followed by Multiplication/Division, and then Addition/Subtraction.

By understanding these steps, evaluating functions becomes an accessible task for solving various problems.

algebraic expressions

Algebraic expressions are combinations of numbers, variables, and operations that represent a specific value or relationship. In the context of evaluating functions, algebraic expressions are used to represent the function itself and to manipulate values.
For example, \( y = -x - 12.1 \) is an algebraic expression where:

• \( -x \) is a negative coefficient variable indicating inverse relationship
• \( -12.1 \) is a constant indicating the starting point or shift in the graph line.

The manipulation of these expressions involves substituting values for variables and simplifying terms. This process allows us to explore function behaviors under different conditions.

Understanding these key components helps in tackling more complex algebra problems, emphasizing that every part of the expression plays a crucial role in defining the function's character and output.