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In the world of linear functions, the slope plays a crucial role in determining how a line behaves. The slope essentially tells us just how steep a line is. To calculate the slope, you need to look at how much the function value (\( f(x) \)) changes when the input value (\( x \)) changes.

For example, in the table given, as \( x \) increases by 2 (from 2 to 4, or from 4 to 6), \( f(x) \) increases by 20 across each interval (from -4 to 16, 16 to 36, etc.). This means the change in \( f(x) \) is consistent.

• Change in \( y \) (\( \Delta y \)) = 20
• Change in \( x \) (\( \Delta x \)) = 2

The slope \( m \) is calculated as:\[m = \frac{\Delta y}{\Delta x} = \frac{20}{2} = 10 \]This slope of 10 tells us that for every step of 1 unit along the \( x \)-axis, the \( y \)-value increases by 10 units.

Once we know the slope, the next step is to find the equation of the line. A handy tool here is the point-slope form of a linear equation. In its simplest terms, this formula combines a known point on the line and its slope to help generate the full equation.

The point-slope form is represented as:\[ y - y_1 = m(x - x_1) \]Where \( m \) is the slope, and \( (x_1, y_1) \) is any point on the line.

• Slope (\( m \)) = 10
• Point \( (x_1, y_1) \) = (2, -4)

Substituting these values into the formula, we have:\[ y + 4 = 10(x - 2) \]Expanding this equation, we solve for \( y \):\[ y + 4 = 10x - 20 \]\[ y = 10x - 24 \]This gives a linear equation in the slope-intercept form of \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept.

Verifying a linear equation involves testing it against known points. This confirms the accuracy of your equation. Once we derived our equation \( y = 10x - 24 \), we substitute the original points to ensure consistency. If all points fit, the equation is correct.

Substitution checks:

• For \( x = 2 \): \( y = 10 \times 2 - 24 = -4 \) (matches)
• For \( x = 4 \): \( y = 10 \times 4 - 24 = 16 \) (matches)
• For \( x = 6 \): \( y = 10 \times 6 - 24 = 36 \) (matches)
• For \( x = 8 \): \( y = 10 \times 8 - 24 = 56 \) (matches)

Since each computed \( y \)-value matches the original table’s \( f(x) \), we've successfully verified that \( y = 10x - 24 \) is indeed the rightful linear equation modeling our data.