Let's analyze each option based on the characteristics provided:
-
Initial value of 6:
- This refers to the value of the function or sequence when \( n = 1 \) (or \( x = 0 \)).
-
y-intercept of 7:
- This is the value of the function when \( x = 0 \).
-
Rate of change (common difference) of 6:
- This characteristic indicates that the function or sequence increases by 6 for each incremental step in \( n \).
-
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:
-
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.
-
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 \( (c) \).
-
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.