Problem 59 - Algebra 1

Question

EVALUATING FUNCTIONS Evaluate the function for these values of \(x:-2,-1,0,\) and \(1 .\) Organize your results in a table. $$y=x-8$$

Step-by-Step Solution

Answer

The values for \(y\) when \(x=-2, x=-1, x=0\) and \(x=1\) are -10, -9, -8 and -7 respectively.

Step 1: Substitute \(x=-2\) into \(y=x-8\)

To find the corresponding \(y\) value when \(x=-2\), substitute \(x=-2\) into the function \(y=x-8\). So it becomes \(y=(-2)-8 = -10\).

Step 2: Substitute \(x=-1\) into \(y=x-8\)

To find the corresponding \(y\) value when \(x=-1\), substitute \(x=-1\) into the function \(y=x-8\). So it becomes \(y=(-1)-8 = -9\).

Step 3: Substitute \(x=0\) into \(y=x-8\)

To find the corresponding \(y\) value when \(x=0\), substitute \(x=0\) into the function \(y=x-8\). So it becomes \(y=(0)-8 = -8\).

Step 4: Substitute \(x=1\) into \(y=x-8\)

To find the corresponding \(y\) value when \(x=1\), substitute \(x=1\) into the function \(y=x-8\). So it becomes \(y=(1)-8 = -7\).

Key Concepts

Algebraic Functions

An algebraic function is a type of function that can be represented using algebraic operations, such as addition, subtraction, multiplication, division, and exponentiation by a constant. The expression involves only algebraic symbols and operations. In the context of our exercise,

y = x - 8
is a simple algebraic function comprising a variable (x), constants, and the fundamental operations of subtraction. Such functions are foundational in algebra as they set the stage for understanding how variables and constants interact within an equation. They are also the building blocks for more complex functions used across various fields of study.

The Substitution Method

The substitution method is a fundamental technique in algebra used to evaluate functions. It involves replacing the variable in a function with a specific value. When you substitute a given number for the variable in the function, you can solve for the corresponding output.

In practice, it's like telling a story: for instance, if 'x' is a placeholder for a character in our story, by substiting 'x' with a specific character, we can see how the narrative unfolds. In terms of mathematics, this process allows us to understand the behavior of the function at particular points, which is essential in graphing and in solving equations where the function equals a certain value.

Function Evaluation

Function evaluation is the process of finding the output of a function for a specific input. The key to understanding this concept lies in matching each input, or 'x', with its correct output, 'y'.

To evaluate a function, simply replace the variable in the function's formula with the given input value and perform the algebraic operations as indicated. The exercise at hand demonstrates this beautifully by asking us to evaluate the function for several values of 'x'. The clarity of function evaluation is crucial, as it is a tool used incessantly in mathematics, from simple calculations to modeling real-world scenarios.

Linear Equations

Linear equations are the simplest type of algebraic function and are represented by the formula of a straight line, typically written as

y = mx + b
where 'm' is the slope of the line and 'b' is the y-intercept, the point at which the line crosses the y-axis. In linear equations, 'x' and 'y' have a direct relationship; when 'x' increases or decreases, 'y' will increase or decrease in a predictable way, determined by the slope 'm'.

Our exercise features a linear equation with a slope of 1 and a y-intercept of -8. This signifies that for each unit increase in 'x', 'y' will increase by 1 unit, and when 'x' is 0, 'y' will be -8. Understanding linear equations is key to grasping more complex algebraic concepts and is often where we begin our graphing journey.