Find the equation of the linear function represented by the table below in slope-intercept form.
$x$ | $y$ |
1 | 4 |
2 | 6 |
3 | 8 |
4 | 10 |
We are given a linear relationship between \(x\) and \(y\) represented by the following table:
| \(x\) | \(y\) \) \(1\) | \(4\) \) \(2\) | \(6\) \) \(3\) | \(8\) \) \(4\) | \(10\) \) |
A linear function follows the slope-intercept form equation \(y = mx + b\), where \(m\) represents the slope (rate of change) and \(b\) represents the \(y\)-intercept. Our goal is to determine the values of \(m\) and \(b\) to define this specific linear function.
To find the slope \(m\), we use any two points \((x_1, y_1)\) and \((x_2, y_2)\) from the table, such as \((1, 4)\) and \((2, 6)\). The formula for the slope is:
\[m = \frac{y_2 - y_1}{x_2 - x_1}\]
Substituting our points:
\[m = \frac{6 - 4}{2 - 1} = \frac{2}{1} = 2\]
This calculation shows that for every increase of 1 in \(x\), \(y\) increases by 2. We can verify this with other points: for \((3, 8)\) and \((4, 10)\), the slope is \(\frac{10 - 8}{4 - 3} = \frac{2}{1} = 2\). The slope is consistent throughout the data, confirming the function is indeed linear.
Now that we know \(m = 2\), our equation looks like \(y = 2x + b\). To find the \(y\)-intercept \(b\), we substitute the coordinates of any point from the table into this equation. Using the point \((1, 4)\):
\[4 = 2(1) + b\]
\[4 = 2 + b\]
Subtracting 2 from both sides gives:
\[b = 2\]
So, the linear equation is \(y = 2x + 2\). We can verify this with another point, such as \((4, 10)\):
\[10 = 2(4) + 2 = 8 + 2 = 10\]
The equation is consistent with all given data points.
Linear functions are used in many real-world situations, such as calculating the cost of a ride-sharing service. If a service charges a base fee of \(\$2\) (the y-intercept) and an additional \(\$2\) for every kilometer traveled (the slope), the total cost \(y\) for traveling \(x\) kilometers is given by \(y = 2x + 2\). This formula allows the user to quickly estimate the fare for any distance.