Problem 12 - Algebra 1: Concepts and Skills

Question

Make an input-output table for the function. Use 0, 1, 2, 3, 4, and 5 as values for x. $$ y=2(6 x+10) $$

Step-by-Step Solution

Answer

The input-output table for the function \( y=2(6x+10) \) is: \n\n| x | y |\n| 0 | 20 |\n| 1 | 32 |\n| 2 | 44 |\n| 3 | 56 |\n| 4 | 68 |\n| 5 | 80 |

Step 1: Identify the Task & Variables

The task is to create an input-output table for the function \( y=2(6x+10) \). The values for x given are 0, 1, 2, 3, 4, and 5.

Step 2: Finding outputs (y) for inputs (x)

To find y for each x, substitute each given x value into the equation and solve for y. Doing this for the given x values results in the following pairs (x, y)

Step 3: Create Table

The final step is to combine the results into an input-output table. This will be a 2-column table with 'x' and 'y' as headers, and the corresponding values in each row.

Key Concepts

Input-Output Tables

An input-output table is a simple way to organize values and results in functions. It helps us understand how changing one variable affects another. In algebra, when we're given a function, an input-output table allows us to see the outcomes based on different inputs. For the function provided in the exercise, \(y = 2(6x + 10)\), we're looking to observe how values of \(x\) influence \(y\).

Here's how we deal with input-output tables:

• Write down the input values for the variable \(x\), in this case, 0 through 5.
• Substitute each value into the function to find the corresponding output \(y\).
• Record the pairs \((x, y)\) in the table.

So, for example, if \(x = 0\), plug this into the function:

• \(y = 2(6(0) + 10) = 2 \times 10 = 20\)

This means the first row in our table will be (0, 20). By repeating this process for given \(x\)-values, we complete our table, seeing how each \(x\) relates to a corresponding \(y\). This is a wonderful way to visualize how input translates to output.

Algebraic Expressions

Algebraic expressions form the backbone of algebra. They consist of numbers, variables (like \(x\)), and operations (such as addition or multiplication). In the exercise above, the expression \(2(6x + 10)\) combines these elements to represent a mathematical relationship.

Breaking it down:

• The expression inside the parentheses, \(6x + 10\), features a variable part \(6x\) and a constant part \(10\).
• The whole expression is multiplied by 2, which affects the output \(y\) directly.

Understanding algebraic expressions involves:

• Identifying numeric coefficients and constant terms.
• Recognizing operations that connect different parts of the expression.

When working with such expressions, it's crucial to follow these steps for accurate results. Each component has a role in defining the expression's value for any given input of \(x\). Mastery of algebraic expressions is key to success in higher-level mathematics.

Solving Equations

Solving equations is a core skill in algebra. It involves finding the values of variables that make the equation true. In the context of the exercise, solving the equation \(y = 2(6x + 10)\) for each given value of \(x\) is what creates our input-output table.

Here are the basic steps to solve for \(y\):

• Substitute the \(x\) value into the equation.
• Perform arithmetic operations to simplify the expression and solve for \(y\).

For example, substituting \(x = 3\), we solve:

• First, calculate inside the parentheses: \(6 \times 3 + 10 = 18 + 10 = 28\).
• Then multiply by 2: \(y = 2 \times 28 = 56\).

Continue this method for each \(x\)-value to understand how the expression simplifies and solve for \(y\). Practice builds confidence in solving equations efficiently.