Which Linear Function Represents A Slope Of 1 4 \frac{1}{4} 4 1 ​ ?${ \begin{array}{|c|c|} \hline x & Y \ \hline 3 & -11 \ \hline 6 & 1 \ \hline 9 & 13 \ \hline 12 & 25 \ \hline \end{array} }$

Which Linear Function Represents a Slope of $\frac{1}{4}$?

In mathematics, a linear function is a function that can be written in the form $y = mx + b$, where $m$ is the slope and $b$ is the y-intercept. The slope of a linear function represents the rate of change of the function with respect to the input variable. In this article, we will explore which linear function represents a slope of $\frac{1}{4}$.

Understanding the Problem

The problem provides a table of values for a linear function, with $x$ and $y$ values. We are asked to find the linear function that represents a slope of $\frac{1}{4}$. To solve this problem, we need to use the concept of slope and the equation of a linear function.

Recall the Equation of a Linear Function

The equation of a linear function is given by $y = mx + b$, where $m$ is the slope and $b$ is the y-intercept. The slope of a linear function represents the rate of change of the function with respect to the input variable.

Finding the Slope

To find the slope of the linear function represented by the table of values, we can use the formula:

$$m = \frac{y_2 - y_1}{x_2 - x_1}$$

where $(x_1, y_1)$ and $(x_2, y_2)$ are two points on the line.

Using the Table of Values

We can use the table of values to find the slope of the linear function. Let's choose two points from the table: $(3, -11)$ and $(6, 1)$. We can plug these values into the formula for the slope:

$$m = \frac{1 - (-11)}{6 - 3} = \frac{12}{3} = 4$$

However, we are looking for a slope of $\frac{1}{4}$, not $4$. This means that the linear function represented by the table of values is not the one we are looking for.

Finding the Correct Linear Function

To find the correct linear function, we need to find the equation of the line that passes through the point $(3, -11)$ and has a slope of $\frac{1}{4}$. We can use the point-slope form of the equation of a line:

$$y - y_1 = m(x - x_1)$$

where $(x_1, y_1)$ is a point on the line and $m$ is the slope.

Using the Point-Slope Form

We can plug in the values $(3, -11)$ and $m = \frac{1}{4}$ into the point-slope form:

$$y - (-11) = \frac{1}{4}(x - 3)$$

Simplifying the equation, we get:

$$y + 11 = \frac{1}{4}x - \frac{3}{4}$$

Multiplying both sides by $4$ to eliminate the fraction, we get:

$$4y + 44 = x - 3$$

Rearranging the equation to put it in the form $y = mx + b$, we get:

$$y = -\frac{1}{4}x + \frac{47}{4}$$

In this article, we explored which linear function represents a slope of $\frac{1}{4}$. We used the concept of slope and the equation of a linear function to find the correct linear function. The equation of the linear function is $y = -\frac{1}{4}x + \frac{47}{4}$.

The final answer is $y = -\frac{1}{4}x + \frac{47}{4}$.
Q&A: Which Linear Function Represents a Slope of $\frac{1}{4}$?

In our previous article, we explored which linear function represents a slope of $\frac{1}{4}$. We used the concept of slope and the equation of a linear function to find the correct linear function. In this article, we will answer some frequently asked questions related to the topic.

Q: What is the slope of a linear function?

A: The slope of a linear function represents the rate of change of the function with respect to the input variable. It is a measure of how much the output changes when the input changes by one unit.

Q: How do I find the slope of a linear function?

A: To find the slope of a linear function, you can use the formula:

$$m = \frac{y_2 - y_1}{x_2 - x_1}$$

where $(x_1, y_1)$ and $(x_2, y_2)$ are two points on the line.

Q: What is the equation of a linear function?

A: The equation of a linear function is given by $y = mx + b$, where $m$ is the slope and $b$ is the y-intercept.

Q: How do I find the y-intercept of a linear function?

A: To find the y-intercept of a linear function, you can use the equation $y = mx + b$. The y-intercept is the value of $b$ when $x = 0$.

Q: What is the point-slope form of the equation of a line?

A: The point-slope form of the equation of a line is given by:

$$y - y_1 = m(x - x_1)$$

where $(x_1, y_1)$ is a point on the line and $m$ is the slope.

Q: How do I use the point-slope form to find the equation of a line?

A: To use the point-slope form to find the equation of a line, you can plug in the values of $(x_1, y_1)$ and $m$ into the equation. Then, simplify the equation to put it in the form $y = mx + b$.

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

A: A linear function is a function that can be written in the form $y = mx + b$, where $m$ is the slope and $b$ is the y-intercept. A non-linear function is a function that cannot be written in this form.

Q: How do I determine if a function is linear or non-linear?

A: To determine if a function is linear or non-linear, you can try to write it in the form $y = mx + b$. If you can write it in this form, then it is a linear function. If you cannot write it in this form, then it is a non-linear function.

In this article, we answered some frequently asked questions related to the topic of which linear function represents a slope of $\frac{1}{4}$. We hope that this article has been helpful in clarifying any confusion you may have had about the topic.

The final answer is that the equation of the linear function that represents a slope of $\frac{1}{4}$ is $y = -\frac{1}{4}x + \frac{47}{4}$.