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A slope is fundamental in understanding linear functions as it indicates the rate of change. In simpler terms, the slope tells you how steep a line is. Whenever the values of a linear function are represented on a graph, the slope dictates how the line moves as you go from one point to another. For instance, in our original scenario where we have the table with values, you can notice that as the value of \(x\) increases from \(-1\) to \(0\) and then to \(1\), the value of \(y\) progresses. The formula for slope, \(m\), is seen in the equation of a line, which is in the form of \(y = mx + b\). Here, \(m\) represents the slope. To find the slope between two points, the formula used is:\[m = \frac{y_2 - y_1}{x_2 - x_1} \]Using the points \((-1,5)\) and \((0,8)\), the change in \(y\) is \(3\) (from \(5\) to \(8\)), while the change in \(x\) is \(1\) (from \(-1\) to \(0\)). This gives us \(m = \frac{3}{1} = 3\). This slope of \(3\) tells us that for every increase of \(1\) in \(x\), \(y\) increases by \(3\). This kind of understanding helps in predicting values and analyzing the behavior of the function.

Filling in missing values in a function table requires understanding the pattern or relationship that exists between the values. In this context, for a linear function, once you have the equation of the line, it becomes straightforward to estimate any unknown values in the function table.With the linear function already determined as \(y = 3x + 8\) from our previous calculations, estimating the missing value for \(y\) when \(x = 1\) involves substituting the \(x\) value into our function. Plugging in \(x = 1\), the calculation becomes:

• \( y = 3(1) + 8 \)
• \( y = 3 + 8 \)
• \( y = 11 \)

Thus, the missing value for \(y\) is \(11\). Understanding this substitution process is crucial as it allows you to estimate and fill in missing numbers for any given \(x\), utilizing the linear relationship you've discovered.

A function table is an easy way of organizing the values of a function and understanding its linear relationship. The table presents a series of \(x\) and \(y\) value pairs that illustrate the function's rate of change, also known as the slope.The role of the function table here is to present known values while marking the unknowns for computation. Looking at our completed table:

• \(x = -1, y = 5\)
• \(x = 0, y = 8\)
• \(x = 1, y = 11\)

Interpreting such a function table allows for identifying the linear pattern—which, in this case, is the increase of \(y\) as \(x\) increases.Reading across a table filled with these values, one can easily understand how changing \(x\) affects \(y\), framing a constant rate of change which defines a linear relationship. The table acts as a simple visual cue to predict missing values by extending the line of inquiry already established by observed values and a calculated slope. This also facilitates a quick understanding of linear functions to those new to these mathematical concepts.