The table represents a linear function.
$x$ | $y$ |
-2 | -2 |
-1 | 1 |
0 | 4 |
1 | 7 |
2 | 10 |
The given data provides a set of coordinate pairs \((x, y)\) that follow a linear relationship. A linear function can be represented by the equation \(y = mx + b\), where \(m\) is the slope of the line and \(b\) is the y-intercept. The slope \(m\) represents the constant rate of change of \(y\) with respect to \(x\). To determine the slope, we can choose any two distinct points \((x_1, y_1)\) and \((x_2, y_2)\) from the provided table and use the slope formula:
\[m = \frac{y_2 - y_1}{x_2 - x_1}\]
The points provided in the table are \((-2, -2), (-1, 1), (0, 4), (1, 7),\) and \((2, 10)\).
To find the slope, we select two points from the table. Let us use \((x_1, y_1) = (-2, -2)\) and \((x_2, y_2) = (-1, 1)\). Substituting these values into the slope formula gives:
\[m = \frac{1 - (-2)}{-1 - (-2)}\]
Simplifying the expression:
\[m = \frac{1 + 2}{-1 + 2} = \frac{3}{1} = 3\]
Thus, the slope is \(3\). To confirm this result, we can select two other points, such as \((0, 4)\) and \((1, 7)\):
\[m = \frac{7 - 4}{1 - 0} = \frac{3}{1} = 3\]
The slope remains consistent across the provided points, confirming that the slope of the linear function is \(3\).
We have determined that the slope of the linear function is \(3\). Comparing this to the given options (\(-3, -2, 3, 4\)), the correct value is \(3\). Therefore, the slope of the function is \(\boxed{3}\).
The concept of slope is essential in understanding real-world rates of change. For example, if you are saving money at a constant rate, the amount of money you have over time can be represented by a linear function. If you start with \$40 and add \$30 every week, the relationship between your total savings \(y\) and the number of weeks \(x\) would be \(y = 30x + 40\). Here, the slope is 30, representing your weekly savings rate. Calculating the slope helps predict future outcomes and analyze trends in various fields like economics, physics, and daily personal finance.