Problem 62 - Algebra 1

Question

EVALUATING FUNCTIONS Evaluate the function for these values of \(x:-2,-1,0,\) and \(1 .\) Organize your results in a table. $$ y=-8.5-(-x) $$

Step-by-Step Solution

Answer

The function \(y = -8.5 + x\) evaluated for \(x = -2, -1, 0, 1\) yields \(y = -10.5, -9.5, -8.5, -7.5\) respectively, when organized in a table.

Step 1: Understand the function

The function is given as \(y = -8.5 - (-x)\), which simplifies to \(y = -8.5 + x \), because two negatives make a positive.

Step 2: Substitute x-values into the function

Substitute the given x-values into the simplified function one by one, and solve for corresponding y-values: \nFor \(x = -2\), \(y = -8.5 + (-2) = -10.5\),\nFor \(x = -1\), \(y = -8.5 + (-1) = -9.5\),\nFor \(x = 0\), \(y = -8.5 + 0 = -8.5\),\nFor \(x = 1\), \(y = -8.5 + 1 = -7.5\) .

Step 3: Organize results in a table

Write the x-values and their corresponding y-values in table form, thus: \n \[\begin{tabular}{||c c||} \hlinex & y \ [0.5ex] \hline\hline-2 & -10.5 \ \hline-1 & -9.5 \\hline0 & -8.5 \\hline1 & -7.5 \\hline\end{tabular}\]

Key Concepts

Algebraic Functions

An algebraic function is like a cooking recipe for numbers. It uses a variety of mathematical operations—like addition, subtraction, multiplication, and division—to take an input value, often referred to as 'x', and produces an output, often labeled 'y'. The expression that describes the relationship between 'x' and 'y' is known as the function's rule.

Just as different recipes yield different dishes, diverse algebraic functions describe different patterns of relationships between variables. For example, the function in our exercise, \(y = -8.5 + x\), can be visualized as a straight line on a graph, tilting upwards as you move from left to right. Each input value of 'x' leads to a unique output 'y', showing how each variable affects the other.

Function Tables

Imagine a function table as a ledger that keeps a neat record of a function's inputs and outputs. Much like a journal tracks daily activities, a function table lists down all the specific values that 'x' can take and the resulting 'y' values after applying the function's rule. What makes it so handy? It allows you to quickly check any given 'x' value and find out what 'y' corresponds to it.

For our problem, a function table aids in organizing the results systematically, reducing confusion and errors. By laying out the inputs '-2, -1, 0, 1' and applying the function rule, we can visually match each input with its output, creating a clear roadmap of the function's behavior across different values.

Solving Functions

Solving functions is akin to decoding a secret message. Each time you're given an 'x', your mission is to figure out what 'y' is by using the function's rule. It's essential to comprehend the rule fully since this is your decoder tool. Here's your mission for the provided exercise:

• Understand the rule: \(y = -8.5 - (-x)= -8.5 + x\), two negatives give a positive.
• Decrypt each 'x': Substitute it into the rule to get 'y'.
• Record your findings in the function table for clarity and to prevent any mix-ups.

By substituting the given 'x' values into the function's equation, you uncover the respective 'y'. Each calculated 'y' value is then plotted in the function table, lining up with its paired 'x', which allows clear visualization of how 'x' influences 'y'.