What is the slope of this table?
x | y |
0 | 0 |
2 | 6 |
4 | 12 |
The problem provides a table of values representing a linear relationship between \(x\) and \(y\). The points given are \((0, 0)\), \((2, 6)\), and \((4, 12)\). We need to determine the slope \(m\) of the line that passes through these points. The slope of a linear equation in the form \(y = mx + b\) represents the rate of change of \(y\) with respect to \(x\). The formula to calculate the slope using any two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by:
\[m = \frac{y_2 - y_1}{x_2 - x_1}\]
To find the slope, we can choose any two points from the table. Let's select \((x_1, y_1) = (0, 0)\) and \((x_2, y_2) = (2, 6)\). Substituting these values into the slope formula:
\[m = \frac{6 - 0}{2 - 0} = \frac{6}{2} = 3\]
Thus, the calculated slope is 3.
To ensure our result is correct and that the relationship is indeed linear with a constant slope, let's verify using another pair of points from the table. Using \((x_1, y_1) = (2, 6)\) and \((x_2, y_2) = (4, 12)\):
\[m = \frac{12 - 6}{4 - 2} = \frac{6}{2} = 3\]
Since both calculations yield the same value, the slope of the linear relationship defined by the table is 3. This corresponds to option C.
Understanding how to calculate the slope is essential in real-life applications. For example, if you are saving money in a piggy bank at a constant rate, say \$3 each day, the amount of money in your bank over time forms a linear relationship. If you have \$0 on day 0, \$6 on day 2, and \$12 on day 4, the 'slope' of your savings is \$3 per day. Knowing this helps you predict your future savings or calculate how long it will take to reach a certain financial goal.