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The slope of a line in a linear function tells us how steep the line is and the direction in which it inclines or declines. It is represented by the variable \(m\) in the function \(f(x) = 1 + m(x + 3)\). The slope can be positive, negative, or zero, and each of these states has unique characteristics.

• A positive slope means the line rises from left to right.
• A negative slope means the line falls from left to right.
• A slope of zero indicates a horizontal line.

By adjusting the value of \(m\), we can see how altering the slope changes how the line appears on the graph. For instance, a slope of 1 results in an upward diagonal line, and a slope of -1 results in a downward diagonal. Understanding the slope is crucial as it directly affects how linear functions behave and intersect with other lines.

An intersection point is where a line crosses a specific point on the graph. It is a crucial characteristic shared by the entire family of lines in our given linear function. In this case, the intersection point is \((-3, 1)\).Let's see why this happens: by plugging \(x = -3\) into the function \(f(x) = 1 + m(x + 3)\), we get \(f(-3) = 1 + m(0) = 1\). This means that for every line within this family, no matter the value of \(m\), the function value is always \(1\) at \(x = -3\).So, all lines in this family intersect the vertical line \(x = -3\) at the same point \((-3, 1)\). Recognizing intersection points like this helps us understand how different lines might interact or align on a graph.

To graph a line from a linear function, we need to understand both its slope and a point it passes through. Here, we can use the slope-intercept form easily for sketching. For instance, if we have the equation \(f(x) = x + 4\), the slope is \(1\), and it has a y-intercept at \(y = 4\).In our family of functions \(f(x) = 1 + m(x+3)\), we start by identifying that all lines pass through \((-3, 1)\). Next, choose a value for \(m\) to determine the slope, such as \(m = 2\). This gives us a steeper line that rises faster.Sketch several lines by:

• Finding the intersection point at \((-3, 1)\).
• Applying the slope \(m\) to determine the rise over run.
• Drawing lines for different values of \(m\).

Use these steps to see how changes in \(m\) create different lines which all meet at the common point \((-3, 1)\). Graphing these lines helps visually demonstrate linear relationships and family characteristics.