Problem 10 - 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. $$ y=85-15 x $$

Step-by-Step Solution

Answer

The input-output pairs for the function \(y = 85 - 15x\) are: (0, 85), (1, 70), (2, 55), (3, 40), (4, 25), (5, 10).

Step 1: Understand the function

Our function here is \(y = 85 - 15x\). It's a linear function where x is the input and y is the output.

Step 2: Start with x = 0

Put x = 0 into the function we get: \( y = 85 - 15*0 = 85 \). So, when x is 0, the output (y) is 85.

Step 3: Move to x = 1

Next, let's put x = 1 into the function. We find: \( y = 85 - 15*1 = 70\). So, when x is 1, y is 70.

Step 4: Continue in the same manner

We keep substituting the given x values into the function to get the corresponding y values. When x = 2, \( y = 85 - 15*2 = 55\). When x = 3, \( y = 85 - 15*3 = 40\). When x = 4, \(y = 85 - 15*4 = 25\). Finally, when x = 5, \(y = 85 - 15*5 = 10\).

Step 5: Wrapping up

After substituting all x values into the function, we have the pairings: (x, y) = {(0, 85), (1, 70), (2, 55), (3, 40), (4, 25), (5, 10)}. This completes our input-output table for the function.

Key Concepts

Input-Output Table

An input-output table is a fundamental tool that helps illustrate how a function behaves, essentially mapping each input value to its corresponding output value. Such tables are often used in mathematics to visualize the relationship between variables in a function. For the linear function given by the formula \( y = 85 - 15x \), we create the table by choosing a set of input values for \( x \), often starting with zero and then moving to consecutive integers. The result for each input is calculated and noted as the output, creating pairs that show how changing the input \( x \) affects the output \( y \).

For students, it's crucial to remember that an input-output table isn't just a chart to fill out but a representation of a consistent mathematical relationship. Creating a table from a function involves not only systematic calculation but also the need to understand the underlying concept of how each 'x' influences 'y' in the given equation.

Function Evaluation

Evaluating a function is the process of determining the output value for a particular input. In the context of our exercise, to evaluate the linear function \( y = 85 - 15x \), we substitute specific values for \( x \) and perform the indicated operations. Working through each given input value—0, 1, 2, 3, 4, and 5—we follow a systematic approach, putting each into the function to get the corresponding \( y \) value.

For clearer understanding, remember that evaluating a function is analogous to following a recipe. Each 'x' value is like an ingredient, and the function itself is the recipe; when we 'cook' them together, the 'y' value is the result we get. Function evaluation is one of the foundational skills needed to work with functions effectively.

Algebraic Expressions

An algebraic expression is a mathematical statement that includes numbers, variables, and arithmetic operations. In our exercise, \( y = 85 - 15x \) is an algebraic expression that defines a linear function, indicating a direct relationship between 'y' (the dependent variable) and 'x' (the independent variable). Understanding such expressions is critical because they are the building blocks of the algebraic functions we encounter.

The simplicity or complexity of algebraic expressions can vary widely, but their purpose remains to express the relationship between variables concisely. By manipulating these expressions—adding, subtracting, multiplying, or dividing—we can solve for unknowns, create models, and explore various mathematical relationships.

Substituting Values

Substituting values in algebra involves replacing variables with actual numbers to evaluate expressions or solve equations. It's a fundamental technique used to find out what the output of a function is for a given input. Referring back to our function \( y = 85 - 15x \), when we 'substitute' numbers for 'x', we're essentially seeing how different values of 'x' change the outcome 'y'.

To correctly substitute a value, ensure that the replacement is done consistently throughout the expression and that the standard order of operations is followed. This concept extends beyond just plug-and-chug calculations; it's a practical application of understanding how variables interact within an equation and how changing one element can influence the entire expression.