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The slope of a linear function is a crucial concept in understanding how the function behaves. It tells us how much the value of the function (or the output "y") will change with each one-unit increase in "x" (the input). In simpler terms, if you think of a graph of the function as a hill, the slope represents how steep the hill is.

To find the slope, we can use the formula:

• \( m = \frac{y_2 - y_1}{x_2 - x_1} \)

Here, \(m\) is the slope, \(x_1\) and \(y_1\) are the coordinates of one point on the line, and \(x_2\) and \(y_2\) are the coordinates of another point.

In our example, using the points \((5.2, 27.8)\) and \((5.3, 29.2)\), we calculated the slope as \(14\). This tells us that for each increase of 1 in \(x\), \(y\) increases by 14. A steep, positive slope indicates a sharp increase in values.

The y-intercept is the point where the line of a linear function crosses the y-axis. It's an essential part of the linear function, represented by the \(b\) in the equation \(y = mx + b\). This tells you the value of \(y\) when \(x = 0\).

To find the y-intercept, you can rearrange the linear equation to solve for \(b\). Using one of the points, like \((5.2, 27.8)\), and our previously found slope of \(14\), we substituted it into the equation:

• \(27.8 = 14(5.2) + b\)

Solving this gives us \(b = -45\). Thus, the y-intercept of our function is \(-45\), meaning that if the function line extended to where \(x = 0\), \(y\) would be \(-45\).

Remember, the y-intercept can sometimes be positive, negative, or zero, depending on the line's position.

Linear equations describe straight lines on a graph and are in the form \(y = mx + b\). This is a simple yet powerful equation that models relationships in various fields, from science to economics.

Let’s break it down:

• \(m\): The slope, showing how much \(y\) changes as \(x\) changes.
• \(b\): The y-intercept, indicating the function's starting point at \(x = 0\).

In our scenario, the linear equation generated from the table's data is \(y = 14x - 45\). Each part plays a role in depicting how x and y are related.

Understanding linear equations is fundamental, as they help solve real-world problems by predicting outcomes and patterns when variables change linearly.

Verifying that a linear function accurately represents the data is as important as finding the function itself. This step ensures that calculations are correct and the function truly models the data.

Function verification involves plugging in other known values of \(x\) from the data table into the derived function and checking if it yields the corresponding \(y\). Using our function \(y = 14x - 45\), we checked the point where \(x = 5.4\) and found that \(y = 30.6\), which matches the given table value.

• Substitute \(x\) into the equation.
• Calculate \(y\).
• Verify if calculated \(y\) matches the table.

If all other original data points verify, it's likely that the linear function is correct. It serves as a double-check, reinforcing the function's reliability before using it for predictions or analysis.