- Determine the slope \(m\) using two points from the table: \(m = \frac{14 - (-6)}{6 - (-2)} = \frac{5}{2}\).
- Identify the point-slope form formula: \(y - y_1 = m(x - x_1)\).
- Substitute the point \((-2, -6)\) and slope \(\frac{5}{2}\) into the formula.
- The resulting equation is \(\boxed{y + 6 = \frac{5}{2}(x + 2)}\).
Explanation
- Analyzing the Problem and Point-Slope Formula
The table provides a series of points \((x, y)\) that satisfy a linear relationship: \((-4, -11), (-2, -6), (6, 14),\) and \((10, 24)\). We are asked to write the equation of this line in point-slope form using the specific point \((-2, -6)\). The point-slope form of a linear equation is given by the formula:
\[y - y_1 = m(x - x_1)\]
where \(m\) is the slope of the line and \((x_1, y_1)\) is a known point on the line. Given our point \((x_1, y_1) = (-2, -6)\), substituting these values into the formula yields:
\[y - (-6) = m(x - (-2))\]
\[y + 6 = m(x + 2)\]
- Calculating the Slope
To find the slope \(m\) of the line, we can use the formula \(m = \frac{y_2 - y_1}{x_2 - x_1}\) with any two points from the table. Let us use the points \((-2, -6)\) and \((6, 14)\):
\[m = \frac{14 - (-6)}{6 - (-2)} = \frac{14 + 6}{6 + 2} = \frac{20}{8} = \frac{5}{2}\]
We can verify this slope with another pair of points, such as \((-4, -11)\) and \((-2, -6)\):
\[m = \frac{-6 - (-11)}{-2 - (-4)} = \frac{-6 + 11}{-2 + 4} = \frac{5}{2}\]
The slope is consistently \(\frac{5}{2}\).
- Finalizing the Equation
Now that we have the slope \(m = \frac{5}{2}\) and the point \((-2, -6)\), we substitute them into the point-slope equation we derived in Step 1:
\[y + 6 = \frac{5}{2}(x + 2)\]
Comparing this result with the provided options, we see that the correct representation is indeed the one where the coordinates of the point \((-2, -6)\) are substituted with their proper signs. The correct option is \(y + 6 = \frac{5}{2}(x + 2)\).
Examples
In real-life, point-slope form is incredibly useful for modeling trends. For instance, if you are tracking the temperature change in a laboratory experiment, and you know the starting temperature at a specific time (a "point") and the rate at which the temperature is changing (the "slope"), you can easily write an equation to predict the temperature at any other time. This helps scientists and engineers make quick calculations without needing the full table of data.