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Linear functions are often expressed using the slope-intercept form, which is written as \( y = mx + b \). This form is a straightforward way to understand linear relationships. In this equation:

• \( y \) is the dependent variable.
• \( x \) is the independent variable or input value.
• \( m \) is the slope of the line.
• \( b \) is the y-intercept, the value of \( y \) when \( x = 0 \).

The slope \( m \) tells us how much \( y \) changes for a change in \( x \). In other words, it's the rise over run between two points. The y-intercept \( b \) shows where the line crosses the y-axis. It's a critical starting point when graphing a line.

In the given exercise, the linear equation derived is \( y = 4x + 50 \). Here, \( m = 4 \) indicates that for every unit increase in \( x \), \( y \) increases by 4. The y-intercept \( b = 50 \) tells us that when \( x = 0 \), \( y \) equals 50.

Mathematical ratios play a crucial role in determining if a set of data represents a linear function. The key idea is to check if changes in \( y \) are proportional to the changes in \( x \). This is done by computing the ratio of change in \( y \) to change in \( x \) between each pair of points.

• Step 1: Calculate \( Δy \), the difference in \( y \) values.
• Step 2: Calculate \( Δx \), the difference in \( x \) values.
• Step 3: Find \( \frac{Δy}{Δx} \), the ratio for each point pair.

In the exercise, the ratios \( \frac{Δy}{Δx} \) for all data pairs were calculated as follows: \( \frac{8}{2} = 4 \), \( \frac{32}{8} = 4 \), and \( \frac{40}{10} = 4 \). Since these ratios remained constant, it verified the data represents a linear function. This constant ratio equates to the slope of the linear function, confirming uniform growth or decline across the dataset.

Function table analysis is a valuable method for understanding the behavior of functions using discrete data points. By analyzing a table of values, you can determine whether the relationship between variables is linear. Here are the steps needed in this analysis:

• Identify pairs of \( x \) and \( y \) values.
• Examine the changes in \( y \) values compared to the changes in \( x \) values.
• Check if these changes are consistent across the table.

In the provided table, we see it contains values for \( x \) and \( y \) which, upon investigation, show that differences in \( y \) are consistently proportional to differences in \( x \). When performing function table analysis, noticing these consistent steps helps to find the formula directly using the linear function equation \( y = mx + b \). This confirms that the table supports a linear function and aids in deriving its equation through observed data.