Find the equation of the linear function represented by the table below in slope-intercept form.
$x$ | 0 | 1 | 2 | 3 | 4 |
$y$ | 9 | 17 | 25 | 33 | 41 |
We are given a set of points in a table and tasked with finding the equation of the linear function that describes them. The table provides the following pairs \((x, y)\): \((0, 9), (1, 17), (2, 25), (3, 33),\) and \((4, 41)\). A linear function in slope-intercept form is given by the equation \(y = mx + b\), where \(m\) represents the slope and \(b\) represents the y-intercept. To determine the function, we need to find the values of \(m\) and \(b\).
The slope \(m\) of a linear function can be calculated using any two distinct points \((x_1, y_1)\) and \((x_2, y_2)\) from the data set using the formula:
\[m = \frac{y_2 - y_1}{x_2 - x_1}\]
Using the first two points \((0, 9)\) and \((1, 17)\):
\[m = \frac{17 - 9}{1 - 0} = \frac{8}{1} = 8\]
We can verify this constant slope with other points, for instance \((1, 17)\) and \((2, 25)\):
\[m = \frac{25 - 17}{2 - 1} = \frac{8}{1} = 8\]
Since the slope is constant across all points, the function is indeed linear with a slope of \(m = 8\).
The y-intercept \(b\) is the value of \(y\) when \(x = 0\). Looking at the provided table, when \(x = 0\), the value of \(y\) is \(9\). Therefore, the y-intercept \(b\) is \(9\). Alternatively, we could substitute the slope \(m = 8\) and any point \((x, y)\), such as \((1, 17)\), into the equation \(y = mx + b\) and solve for \(b\):
\[17 = 8(1) + b\]
\[17 = 8 + b\]
\[b = 17 - 8 = 9\]
Thus, we confirm that \(b = 9\).
Now that we have identified the slope \(m = 8\) and the y-intercept \(b = 9\), we can substitute these values into the general slope-intercept form \(y = mx + b\). The resulting equation is \(y = 8x + 9\). Therefore, the linear function represented by the table is \(\boxed{y = 8x + 9}\).
Linear functions are common in daily life, such as when calculating total costs based on a fixed rate. For instance, if you pay a flat monthly fee for a service plus a cost per unit of usage, your bill follows a linear pattern. If you pay \$9 for a subscription and \$8 per hour of usage, the total cost \(y\) for \(x\) hours of usage is exactly described by the function \(y = 8x + 9\). This allows you to quickly predict your expenses based on your activity level.