Question

Based on the tables, describe the difference from Linear Function 1 to Linear Function 2.

Linear Function 1

x f(x)

1 1
2 2
3 3
Linear Function 2


x f(x)

1 6
2 7
3 8

Answer 1

To compare Linear Function 1 and Linear Function 2, we can analyze the values of \(f(x)\) based on the given input values of \(x\).

**Linear Function 1:**
- When \(x = 1\), \(f(x) = 1\)
- When \(x = 2\), \(f(x) = 2\)
- When \(x = 3\), \(f(x) = 3\)

This is a linear function of the form \(f(x) = x\). It has a slope of 1 and a y-intercept of 0.

**Linear Function 2:**
- When \(x = 1\), \(f(x) = 6\)
- When \(x = 2\), \(f(x) = 7\)
- When \(x = 3\), \(f(x) = 8\)

This is a linear function of the form \(f(x) = x + 5\). It also has a slope of 1 but a y-intercept of 5.

**Differences:**
1. **Y-Intercept:** Linear Function 1 has a y-intercept of 0, while Linear Function 2 has a y-intercept of 5. This means that Linear Function 2 is shifted upward by 5 units compared to Linear Function 1.

2. **Value of Output:** The output values \(f(x)\) for Linear Function 1 are all 5 units lower than the corresponding values of Linear Function 2. For example:
- At \(x = 1\), \(f(1) = 1\) (Function 1) vs. \(f(1) = 6\) (Function 2).
- At \(x = 2\), \(f(2) = 2\) (Function 1) vs. \(f(2) = 7\) (Function 2).
- At \(x = 3\), \(f(3) = 3\) (Function 1) vs. \(f(3) = 8\) (Function 2).

3. **Overall Relationship:** Both functions are increasing at the same rate (slope of 1), but no matter the value of \(x\), the output of Linear Function 2 is consistently 5 units greater than that of Linear Function 1.

In summary, the main difference between Linear Function 1 and Linear Function 2 is that Function 2 is a vertical translation of Function 1 upwards by 5 units, while retaining the same slope.