Understanding the Function Rule

Oct 13, 2025 by miodeya.com 92 views

Completing the Remainder of the Table for the Given Function Rule

In mathematics, a function rule is a mathematical expression that describes the relationship between the input and output of a function. Given a function rule, we can use it to find the output value for a specific input value. In this article, we will focus on completing the remainder of the table for the given function rule: $y = 4 - \frac{3x}{4}$ .

Understanding the Function Rule

The given function rule is $y = 4 - \frac{3x}{4}$ . This means that for every input value of $x$ , we can find the corresponding output value of $y$ by plugging in the value of $x$ into the function rule. To complete the remainder of the table, we need to find the output values of $y$ for the input values of $x$ that are not already given.

Completing the Table

To complete the table, we will substitute each input value of $x$ into the function rule and calculate the corresponding output value of $y$ .

Substituting x = 4 into the Function Rule

To find the output value of $y$ for $x = 4$ , we substitute $x = 4$ into the function rule:

$y = 4 - \frac{3(4)}{4}$

Simplifying the expression, we get:

$y = 4 - 3$

$y = 1$

So, the output value of $y$ for $x = 4$ is 1.

Substituting x = 8 into the Function Rule

To find the output value of $y$ for $x = 8$ , we substitute $x = 8$ into the function rule:

$y = 4 - \frac{3(8)}{4}$

Simplifying the expression, we get:

$y = 4 - 6$

$y = -2$

So, the output value of $y$ for $x = 8$ is -2.

Substituting x = 12 into the Function Rule

To find the output value of $y$ for $x = 12$ , we substitute $x = 12$ into the function rule:

$y = 4 - \frac{3(12)}{4}$

Simplifying the expression, we get:

$y = 4 - 9$

$y = -5$

So, the output value of $y$ for $x = 12$ is -5.

In this article, we completed the remainder of the table for the given function rule: $y = 4 - \frac{3x}{4}$ . We substituted each input value of $x$ into the function rule and calculated the corresponding output value of $y$ . The completed table is shown below:

x	y
-4	7
0	4
4	1
8	-2
12	-5

The function rule $y = 4 - \frac{3x}{4}$ is a linear function, which means that it has a constant rate of change. The graph of this function is a straight line with a negative slope. The table shows that as the input value of $x$ increases, the output value of $y$ decreases.

This article covers the following mathematical concepts:

Function rules
Linear functions
Graphing functions
Tables of values

The concept of function rules and linear functions has many real-world applications, such as:

Modeling population growth
Describing the motion of objects
Analyzing data

In conclusion, completing the remainder of the table for the given function rule $y = 4 - \frac{3x}{4}$ requires substituting each input value of $x$ into the function rule and calculating the corresponding output value of $y$ . The completed table shows that as the input value of $x$ increases, the output value of $y$ decreases. This article covers the mathematical concepts of function rules, linear functions, graphing functions, and tables of values, and highlights the real-world applications of these concepts.
Q&A: Completing the Remainder of the Table for the Given Function Rule

In our previous article, we completed the remainder of the table for the given function rule: $y = 4 - \frac{3x}{4}$ . In this article, we will answer some frequently asked questions (FAQs) related to the topic.

Q: What is a function rule?

A: A function rule is a mathematical expression that describes the relationship between the input and output of a function. It is a way to represent a function in a concise and compact form.

Q: What is the difference between a function rule and a table of values?

A: A function rule is a mathematical expression that describes the relationship between the input and output of a function, while a table of values is a list of input and output values of a function. The table of values is a way to represent a function in a tabular form.

Q: How do I find the output value of y for a given input value of x?

A: To find the output value of y for a given input value of x, you need to substitute the value of x into the function rule and simplify the expression.

Q: What is the significance of the slope in a linear function?

A: The slope in a linear function represents the rate of change of the output value with respect to the input value. A positive slope indicates that the output value increases as the input value increases, while a negative slope indicates that the output value decreases as the input value increases.

Q: Can I use a function rule to model real-world phenomena?

A: Yes, you can use a function rule to model real-world phenomena. For example, you can use a linear function to model the growth of a population, the motion of an object, or the cost of a product.

Q: How do I determine if a function is linear or not?

A: To determine if a function is linear or not, you need to check if the function rule is in the form of y = mx + b, where m is the slope and b is the y-intercept.

Q: What is the difference between a linear function and a quadratic function?

A: A linear function is a function that has a constant rate of change, while a quadratic function is a function that has a variable rate of change. A linear function can be represented by a straight line, while a quadratic function can be represented by a parabola.

Q: Can I use a function rule to solve problems in other areas of mathematics?

A: Yes, you can use a function rule to solve problems in other areas of mathematics, such as algebra, geometry, and calculus.

In conclusion, completing the remainder of the table for the given function rule $y = 4 - \frac{3x}{4}$ requires substituting each input value of x into the function rule and calculating the corresponding output value of y. This article answers some frequently asked questions (FAQs) related to the topic, including the definition of a function rule, the difference between a function rule and a table of values, and the significance of the slope in a linear function.

This article covers the following mathematical concepts:

Function rules
Linear functions
Graphing functions
Tables of values
Slope
Real-world applications of functions

The concept of function rules and linear functions has many real-world applications, such as:

Modeling population growth
Describing the motion of objects
Analyzing data
Solving problems in other areas of mathematics

In conclusion, function rules and linear functions are fundamental concepts in mathematics that have many real-world applications. This article provides a comprehensive overview of the topic, including the definition of a function rule, the difference between a function rule and a table of values, and the significance of the slope in a linear function.