Which table shows that y=−f(x) results in a reflection over the x-axis?

Question

(1 point)
Responses

x	−f(x)
−2	0
−3	−1
−6	−2
  x f left parenthesis x right parenthesis negative 2 0 negative 3 negative 1 negative 6 negative 2

x	−f(x)
−2	0
−3	1
−6	2
  x f left parenthesis x right parenthesis negative 2 0 negative 3 1 negative 6 2

x	−f(x)
2	0
3	1
6	2
  x f left parenthesis x right parenthesis 2 0 3 1 6 2

x	−f(x)
2	0
3	−1
6	−2

GPT-4o mini · Answer

To determine which table shows that $ y = -f(x) $ results in a reflection over the x-axis, we need to look for a table where the values of $ -f(x) $ are the negative of the corresponding values of $ f(x) $. Let's analyze each provided table: 1. **First table:** - $ x: -2, -3, -6 $ - $ -f(x): 0, -1, -2 $ - Possible values of $ f(x) $: $ 0, 1, 2 $ - This shows that $ -f(-2) = 0 $, $ -f(-3) = -1 $, $ -f(-6) = -2 $. - This does not represent $ y = -f(x) $ because $ -1 $ and $ -2 $ should be $ 1 $ and $ 2 $ respectively. 2. **Second table:** - $ x: -2, -3, -6 $ - $ -f(x): 0, 1, 2 $ - Possible values of $ f(x) $: $ 0, -1, -2 $ - This shows that $ -f(-2) = 0 $, $ -f(-3) = 1 $, $ -f(-6) = 2 $. - This indicates $ f(-3) = -1 $ and $ f(-6) = -2 $ which is not a correct reflection. 3. **Third table:** - $ x: 2, 3, 6 $ - $ -f(x): 0, 1, 2 $ - Possible values of $ f(x) $: $ 0, -1, -2 $ - This suggests $ f(2) = 0 $, $ f(3) = -1 $, $ f(6) = -2 $. - Values seem inverted, specifically, $ 1 $ should actually represent $ f(3) = -1$ indicating no reflection. 4. **Fourth table:** - $ x: 2, 3, 6 $ - $ -f(x): 0, -1, -2 $ - Possible values of $ f(x) $: $ 0, 1, 2 $ - This indicates that $ -f(2) = 0$, $ -f(3) = -1 $ (hence $ f(3) = 1 $), and $ -f(6) = -2 $ (hence $ f(6) = 2$). - This accurately represents a reflection: Each value of $ -f(x) $ is the negative of its corresponding $ f(x) $. Based on the analysis, the fourth table correctly demonstrates that $ y = -f(x) $ results in a reflection over the x-axis.