Find the equation of the linear function represented by the table below in slope-intercept form.
$x$ | $y$ |
1 | 2 |
2 | 6 |
3 | 10 |
4 | 14 |
The goal is to find the equation of a linear function in the form \(y = mx + b\), where \(m\) represents the slope and \(b\) represents the y-intercept. We are provided with a table of points that satisfy this linear relationship: \((1, 2)\), \((2, 6)\), \((3, 10)\), and \((4, 14)\). A linear function has a constant rate of change, which means the slope between any two points will be the same.
To determine the slope \(m\), we choose any two points from the table, for example \((x_1, y_1) = (1, 2)\) and \((x_2, y_2) = (2, 6)\). The slope formula is given by:
\[m = \frac{y_2 - y_1}{x_2 - x_1}\]
Substituting the chosen points, we get:
\[m = \frac{6 - 2}{2 - 1} = \frac{4}{1} = 4\]
We can verify this with other pairs of points, such as \((3, 10)\) and \((4, 14)\):
\[m = \frac{14 - 10}{4 - 3} = \frac{4}{1} = 4\]
Since the slope is consistent, the slope of our linear function is \(m = 4\).
Now that we have the slope \(m = 4\), we use the slope-intercept form \(y = mx + b\) to find the y-intercept \(b\). We can substitute the slope and any of the points from the table into the equation. Using the point \((1, 2)\):
\[2 = 4(1) + b\]
\[2 = 4 + b\]
Solving for \(b\), we subtract 4 from both sides:
\[b = 2 - 4 = -2\]
Thus, the y-intercept is \(-2\).
With the slope \(m = 4\) and the y-intercept \(b = -2\), we can construct the final equation for the linear function. Substituting these values into \(y = mx + b\), we get:
\[y = 4x - 2\]
This equation describes the relationship between \(x\) and \(y\) for all the points provided in the table. The final answer is \(\boxed{y = 4x - 2}\).
Linear functions are used in many real-world scenarios to represent constant rates of change. For instance, consider a mobile phone plan where you pay a fixed monthly base fee plus a constant amount for every gigabyte of data you use. If the base fee is \$-\$2\$ (perhaps a credit) and you pay \$\$4\$ per gigabyte, the total cost \(y\) for \(x\) gigabytes could be modeled by the equation \(y = 4x - 2\). Understanding these relationships allows us to predict future costs, optimize resource consumption, and make informed financial decisions.