Find the equation of the linear function represented by the table below in slope-intercept form.
$x$ | $y$ |
1 | 6 |
2 | 7 |
3 | 8 |
4 | 9 |
The problem provides a set of points \((x, y)\) representing a linear function: \((1, 6), (2, 7), (3, 8),\) and \((4, 9).\) A linear function is expressed in the slope-intercept form \(y = mx + b,\) where \(m\) represents the slope (the rate of change) and \(b\) represents the y-intercept (the value of \(y\) when \(x = 0\)). Since the function is linear, the slope \(m\) remains constant between any two points.
To calculate the slope \(m\), we use any two points from the table, such as \((x_1, y_1) = (1, 6)\) and \((x_2, y_2) = (2, 7)\). The formula for the slope is:
\[m = \frac{y_2 - y_1}{x_2 - x_1}\]
Substituting the selected points:
\[m = \frac{7 - 6}{2 - 1} = \frac{1}{1} = 1\]
Thus, the slope of the function is \(m = 1\).
Now that we have the slope \(m = 1\), we substitute this and one of the points, for instance \((1, 6)\), into the slope-intercept form \(y = mx + b\) to find the y-intercept \(b\):
\[6 = 1(1) + b\]
Solving for \(b\):
\[6 = 1 + b\]
\[b = 6 - 1 = 5\]
With \(m = 1\) and \(b = 5\), the equation of the linear function is \(y = 1x + 5\), which simplifies to \(y = x + 5\).
To verify the equation \(y = x + 5\), we can test it with the remaining points from the table. For \((3, 8)\), substituting \(x = 3\) gives \(y = 3 + 5 = 8\), which matches the table. For \((4, 9)\), substituting \(x = 4\) gives \(y = 4 + 5 = 9\), which also matches. The equation of the linear function is \(\boxed{y = x + 5}\).
Linear relationships appear in many everyday situations. For example, if you are signing up for a gym membership that costs \(\$5\) for registration plus a \(\$1\) monthly fee, the total cost \(y\) after \(x\) months is represented by the equation \(y = 1x + 5\). By understanding the relationship between the slope and the intercept, you can easily predict your future costs and plan your budget effectively. This simple model helps in understanding how constant rates of change govern many practical financial and physical phenomena.