Question

Match the characteristic to the function that is the same:
a) Initial value of 6
b) y-intercept of 7
c) Rate of change (common difference) of 6
d) Rate of change (common difference) of -4
Options:
an = 6 + 7 (n - 1)
an = -4 + 6 (n - 1)
f (x) = -4x + 2

Answer 1

Let's analyze each option based on the characteristics provided:

1. **Initial value of 6**:
- This refers to the value of the function or sequence when \( n = 1 \) (or \( x = 0 \)).

2. **y-intercept of 7**:
- This is the value of the function when \( x = 0 \).

3. **Rate of change (common difference) of 6**:
- This characteristic indicates that the function or sequence increases by 6 for each incremental step in \( n \).

4. **Rate of change (common difference) of -4**:
- This characteristic indicates that the function or sequence decreases by 4 for each increment in \( n \).

Now, let's match each of the options with the characteristics:

1. **Option**: \( a_n = 6 + 7 (n - 1) \)
- This can be rewritten as \( a_n = 6 + 7n - 7 = 7n - 1 \).
- The y-intercept is \(-1\) when \( n = 1 \), it doesn't match any characteristic.

2. **Option**: \( a_n = -4 + 6 (n - 1) \)
- This can be rewritten as \( a_n = -4 + 6n - 6 = 6n - 10 \).
- The initial value (when \( n = 1 \)) is \( -4 + 6*0 = -4 \).
- The common difference (rate of change) is \( 6 \), which matches \( (c) \).

3. **Option**: \( f(x) = -4x + 2 \)
- The y-intercept is \( 2 \) (when \( x = 0 \)).
- The rate of change (slope) is \(-4\), which matches \( (d) \).

Based on this analysis:

- **(a) Initial value of 6**: Not matched.
- **(b) y-intercept of 7**: Not matched.
- **(c) Rate of change (common difference) of 6**: Matched with \( a_n = -4 + 6(n - 1) \).
- **(d) Rate of change (common difference) of -4**: Matched with \( f(x) = -4x + 2 \).

So, to directly answer your matches, only **(c)** and **(d)** have been matched, and here are the summaries:

- **(c)** with \( a_n = -4 + 6(n - 1) \)
- **(d)** with \( f(x) = -4x + 2 \)

The other options do not match any given characteristic.