Problem 43 - Algebra 1: Concepts and Skills

Question

Evaluate the function when \(x=-2,-1,0\) and \(1 .\) Organize your results in a table. $$ y=12-x $$

Step-by-Step Solution

Answer

The results are \(y=14\) for \(x=-2\), \(y=13\) for \(x=-1\), \(y=12\) for \(x=0\), and \(y=11\) for \(x=1\).

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

Substitute \(x=-2\) into the equation. This gives \(y=12-(-2)=12+2=14\)

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

Substitute \(x=-1\) into the equation. This gives \(y=12-(-1)=12+1=13\)

Step 3: Calculate for \(x = 0\)

Substitute \(x=0\) into the equation. This gives \(y=12-0=12\)

Step 4: Calculate for \(x = 1\)

Substitute \(x=1\) into the equation. This gives \(y=12-1=11\)

Key Concepts

Tables in Mathematics

When working with mathematical functions, tables are invaluable tools for organizing data. Imagine a table as a structured way of displaying input values and their corresponding output values. In our function evaluation example, the function is given by \(y = 12 - x\). By tabulating the results, you can clearly see how changes in \(x\) affect \(y\).

For instance:

• Input \(x = -2\) yields output \(y = 14\)
• Input \(x = -1\) yields output \(y = 13\)
• Input \(x = 0\) results in \(y = 12\)
• Input \(x = 1\) results in \(y = 11\)

Flowing in a sequence, each \(x\) value is clearly connected to its output \(y\). This makes it easier for students to spot patterns, troubleshoot errors, and predict future results. If you grasp the idea behind tables in mathematics, functions become more intuitive and less abstract. Think of tables as a simple roadmap that guides you from a given point to a destination.

Linear Functions

A linear function is a type of algebraic function, defined by an equation that forms a straight line when graphed. The general form of a linear function is \(y = mx + b\), where \(m\) represents the slope and \(b\) the y-intercept. In the given exercise, the function \(y = 12 - x\) is linear with a slope of \(-1\) and a y-intercept of \(12\).

Understanding linear functions means recognizing their set characteristics:

• The constant rate of change: The difference between consecutive \(y\) values is consistent. Here, every time \(x\) increases by 1, \(y\) decreases by 1.
• The straight-line graph: Plotting points from the table on a coordinate system, the points lie on a straight line.
• Simplicity in calculations: Linear equations offer straightforward computations and predictions, making them ideal for quick problem-solving in algebra and calculus.

Linear functions are fundamental, a stepping stone to understanding more complex functions. Their straightforward nature makes them a favorite first step into the world of functions.

Substitution in Algebra

Substitution is a fundamental algebraic technique used to find the value of an unknown in an equation. By replacing variables with actual values, you can solve for unknowns or evaluate expressions. In our specific exercise, substitution involves plugging given \(x\) values into the function \(y = 12 - x\).

Steps for substitution:

• Identify the variables: Here, our variable is \(x\).
• Substitute the value: Input your value of \(x\) into the function. For example, substitute \(x = -2\) to find \(y = 12 - (-2) = 14\).
• Calculate: Perform the arithmetic operation to solve for \(y\).

The art of substitution simplifies seemingly complex problems into manageable calculations. It allows you to methodically evaluate any function or equation, ensuring accuracy and understanding at each step. Mastering substitution empowers you to tackle a wide range of algebraic challenges with confidence.