- The student correctly identifies the slope as \(m = 9\) because the function increases by \(9\) for every \(1\) unit increase in \(x\).
- The student makes an error by assuming the constant term \(b\) is equal to \(f(1)=10\) instead of calculating it for \(x=0\).
- By substituting a point like \((1, 10)\) into the linear form \(f(x) = 9x + b\), we solve \(10 = 9(1) + b\) to find the correct y-intercept \(b = 1\).
- The correct linear function is \(\boxed{f(x) = 9x + 1}\).
Explanation
- Analyzing the Data and Identifying the Slope
To determine the correct function rule, we start by analyzing the given table of data points:
| \(x\) | \(1\) | \(2\) | \(3\) | \(4\) \) \(f(x)\) | \(10\) | \(19\) | \(28\) | \(37\) \) |
The student correctly identified that for every increment of \(1\) in \(x\), the value of \(f(x)\) increases by \(9\). This indicates a constant rate of change, or slope, of \(m = 9\). The general form of a linear function is \(f(x) = mx + b\). Substituting the slope, we have \(f(x) = 9x + b\).
- Identifying and Correcting the Error
The student's error lies in the determination of the y-intercept, \(b\). The student observed that when \(x = 1\), \(f(x) = 10\), and mistakenly assumed that this value corresponds directly to the intercept \(b\) in the expression \(f(x) = mx + b\). However, in a linear function, the y-intercept \(b\) is the value of \(f(x)\) when \(x = 0\), not when \(x = 1\). We can find the correct value of \(b\) by substituting one of the data points into the equation \(f(x) = 9x + b\). Using the point \((1, 10)\):
\[10 = 9(1) + b\]
\[10 = 9 + b\]
\[b = 10 - 9 = 1\]
Thus, the y-intercept is \(b = 1\).
- Concluding and Verifying the Correct Function Rule
Now that we have the correct slope \(m = 9\) and the correct y-intercept \(b = 1\), we can formulate the correct linear function:
\[f(x) = 9x + 1\]
To verify this, we check the rule against the given data points:
For \(x = 1\): \(f(1) = 9(1) + 1 = 10\) (Correct)
For \(x = 2\): \(f(2) = 9(2) + 1 = 19\) (Correct)
For \(x = 3\): \(f(3) = 9(3) + 1 = 28\) (Correct)
For \(x = 4\): \(f(4) = 9(4) + 1 = 37\) (Correct)
The student's error was setting \(b\) equal to \(f(1)\) instead of solving for the intercept at \(x=0\). The correct function rule is \(\boxed{f(x) = 9x + 1}\).
Examples
Linear functions are frequently used in real-world scenarios to model costs that have a fixed base fee plus a variable rate. For example, if you are hiring a plumber, they might charge a flat service fee of \(\$1\) (the intercept) plus \(\$9\) (the rate) for every hour of labor performed. If you have been billed for a job, you can use the function \(f(x) = 9x + 1\) to determine the total cost based on the number of hours (\(x\)) worked. This helps you verify that your bill is accurate based on the time spent on the repair.