Find the equation of the linear function represented by the table below in slope-intercept form.
$x$ | 0 | 1 | 2 | 3 | 4 |
$y$ | 3 | 6 | 9 | 12 | 15 |
We are given a table of values representing a linear function. A linear function can be represented in the form \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. The provided coordinates are (0, 3), (1, 6), (2, 9), (3, 12), and (4, 15). To determine the equation, we need to calculate the constant slope \(m\) and the y-intercept \(b\).
The slope \(m\) of a linear function can be calculated using any two points \((x_1, y_1)\) and \((x_2, y_2)\) from the table using the formula \(m = \frac{y_2 - y_1}{x_2 - x_1}\). Let us use the first two points: \((x_1, y_1) = (0, 3)\) and \((x_2, y_2) = (1, 6)\). Substituting these values into the formula, we get:
\[m = \frac{6 - 3}{1 - 0} = \frac{3}{1} = 3.\]
Thus, the slope \(m\) of the function is \(3\).
The y-intercept \(b\) is the value of \(y\) when \(x = 0\). Looking at the provided table, when \(x = 0\), the corresponding \(y\)-value is \(3\). Therefore, the y-intercept \(b = 3\). Alternatively, we could substitute the point \((1, 6)\) and the slope \(m = 3\) into the equation \(y = mx + b\):
\[6 = 3(1) + b\]
\[6 = 3 + b\]
\[b = 6 - 3 = 3.\]
Both methods confirm that the y-intercept is \(3\).
Now that we have the slope \(m = 3\) and the y-intercept \(b = 3\), we can substitute these values into the general linear equation \(y = mx + b\). This results in the linear function:
\[y = 3x + 3.\]
We can verify this with other points from the table, for example, \(x = 4\): \(y = 3(4) + 3 = 12 + 3 = 15\), which matches the table. The final equation is \(\boxed{y = 3x + 3}\).
Linear functions are frequently used in economics and everyday planning. For example, if a taxi service charges a flat fee of \$3 for pickup (the y-intercept) plus \$3 per mile driven (the slope), the total cost \(y\) for traveling \(x\) miles can be modeled by the linear function \(y = 3x + 3\). Understanding this model helps customers predict costs and businesses set fair pricing strategies.