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To explore the features of the function [tex]\( g(x) \)[/tex] given by [tex]\( g(x) = f(x+4) + 8 \)[/tex] where [tex]\( f(x) = \log_2 x \)[/tex], we will break down the transformation step-by-step.
1. Base Function:
The initial function is [tex]\( f(x) = \log_2 x \)[/tex]. This is a logarithmic function with a base of 2.
2. Horizontal Translation:
The function [tex]\( f(x+4) \)[/tex] represents a horizontal translation of [tex]\( f(x) = \log_2 x \)[/tex] by 4 units to the left. This means for any given input [tex]\( x \)[/tex], we are now evaluating [tex]\( \log_2 (x+4) \)[/tex] instead of [tex]\( \log_2 x \)[/tex].
3. Vertical Translation:
Adding 8 to the result of [tex]\( f(x+4) \)[/tex] translates the entire graph 8 units up. Therefore, [tex]\( g(x) = f(x+4) + 8 \)[/tex] translates to [tex]\( g(x) = \log_2 (x+4) + 8 \)[/tex].
Let's look at the function features step-by-step with some specific values of [tex]\( x \)[/tex]:
### Calculating [tex]\( f(x) = \log_2 x \)[/tex] for specific values of [tex]\( x \)[/tex]:
- For [tex]\( x = 1 \)[/tex], [tex]\( f(1) = \log_2 1 = 0 \)[/tex]
- For [tex]\( x = 2 \)[/tex], [tex]\( f(2) = \log_2 2 = 1 \)[/tex]
- For [tex]\( x = 4 \)[/tex], [tex]\( f(4) = \log_2 4 = 2 \)[/tex]
- For [tex]\( x = 8 \)[/tex], [tex]\( f(8) = \log_2 8 = 3 \)[/tex]
So, we have [tex]\( f(x) = [0.0, 1.0, 2.0, 3.0] \)[/tex].
### Translating the function horizontally by 4 units to the left:
- For [tex]\( x = 1 \)[/tex], [tex]\( f(1+4) = \log_2 (1+4) = \log_2 5 \approx 2.321928094887362 \)[/tex]
- For [tex]\( x = 2 \)[/tex], [tex]\( f(2+4) = \log_2 (2+4) = \log_2 6 \approx 2.584962500721156 \)[/tex]
- For [tex]\( x = 4 \)[/tex], [tex]\( f(4+4) = \log_2 (4+4) = \log_2 8 = 3.0 \)[/tex]
- For [tex]\( x = 8 \)[/tex], [tex]\( f(8+4) = \log_2 (8+4) = \log_2 12 \approx 3.584962500721156 \)[/tex]
Therefore, the horizontally translated values are [tex]\([2.321928094887362, 2.584962500721156, 3.0, 3.584962500721156] \)[/tex].
### Adding 8 to each of these values:
- For [tex]\( x = 1 \)[/tex], [tex]\( g(1) = \log_2 (1+4) + 8 \approx 2.321928094887362 + 8 = 10.321928094887362 \)[/tex]
- For [tex]\( x = 2 \)[/tex], [tex]\( g(2) = \log_2 (2+4) + 8 \approx 2.584962500721156 + 8 = 10.584962500721156 \)[/tex]
- For [tex]\( x = 4 \)[/tex], [tex]\( g(4) = \log_2 (4+4) + 8 = 3.0 + 8 = 11.0 \)[/tex]
- For [tex]\( x = 8 \)[/tex], [tex]\( g(8) = \log_2 (8+4) + 8 \approx 3.584962500721156 + 8 = 11.584962500721156 \)[/tex]
Thus, the final values of [tex]\( g(x) \)[/tex] are [tex]\([10.321928094887362, 10.584962500721156, 11.0, 11.584962500721156] \)[/tex].
### Summary of the transformations and features:
- The function [tex]\( g(x) = \log_2 (x+4) + 8 \)[/tex] is obtained by translating [tex]\( \log_2 x \)[/tex] 4 units to the left and then 8 units up.
- This results in a set of transformations:
- Original function values [tex]\( f(x) \)[/tex] for [tex]\([1, 2, 4, 8] \)[/tex] are [tex]\([0.0, 1.0, 2.0, 3.0] \)[/tex].
- After translating horizontally by 4 units, the values become [tex]\([2.321928094887362, 2.584962500721156, 3.0, 3.584962500721156] \)[/tex].
- After adding 8, the final values of [tex]\( g(x) \)[/tex] are [tex]\([10.321928094887362, 10.584962500721156, 11.0, 11.584962500721156] \)[/tex].
The features of the new function [tex]\( g(x) \)[/tex] include a shifted graph of [tex]\( \log_2 x \)[/tex] both horizontally and vertically.
1. Base Function:
The initial function is [tex]\( f(x) = \log_2 x \)[/tex]. This is a logarithmic function with a base of 2.
2. Horizontal Translation:
The function [tex]\( f(x+4) \)[/tex] represents a horizontal translation of [tex]\( f(x) = \log_2 x \)[/tex] by 4 units to the left. This means for any given input [tex]\( x \)[/tex], we are now evaluating [tex]\( \log_2 (x+4) \)[/tex] instead of [tex]\( \log_2 x \)[/tex].
3. Vertical Translation:
Adding 8 to the result of [tex]\( f(x+4) \)[/tex] translates the entire graph 8 units up. Therefore, [tex]\( g(x) = f(x+4) + 8 \)[/tex] translates to [tex]\( g(x) = \log_2 (x+4) + 8 \)[/tex].
Let's look at the function features step-by-step with some specific values of [tex]\( x \)[/tex]:
### Calculating [tex]\( f(x) = \log_2 x \)[/tex] for specific values of [tex]\( x \)[/tex]:
- For [tex]\( x = 1 \)[/tex], [tex]\( f(1) = \log_2 1 = 0 \)[/tex]
- For [tex]\( x = 2 \)[/tex], [tex]\( f(2) = \log_2 2 = 1 \)[/tex]
- For [tex]\( x = 4 \)[/tex], [tex]\( f(4) = \log_2 4 = 2 \)[/tex]
- For [tex]\( x = 8 \)[/tex], [tex]\( f(8) = \log_2 8 = 3 \)[/tex]
So, we have [tex]\( f(x) = [0.0, 1.0, 2.0, 3.0] \)[/tex].
### Translating the function horizontally by 4 units to the left:
- For [tex]\( x = 1 \)[/tex], [tex]\( f(1+4) = \log_2 (1+4) = \log_2 5 \approx 2.321928094887362 \)[/tex]
- For [tex]\( x = 2 \)[/tex], [tex]\( f(2+4) = \log_2 (2+4) = \log_2 6 \approx 2.584962500721156 \)[/tex]
- For [tex]\( x = 4 \)[/tex], [tex]\( f(4+4) = \log_2 (4+4) = \log_2 8 = 3.0 \)[/tex]
- For [tex]\( x = 8 \)[/tex], [tex]\( f(8+4) = \log_2 (8+4) = \log_2 12 \approx 3.584962500721156 \)[/tex]
Therefore, the horizontally translated values are [tex]\([2.321928094887362, 2.584962500721156, 3.0, 3.584962500721156] \)[/tex].
### Adding 8 to each of these values:
- For [tex]\( x = 1 \)[/tex], [tex]\( g(1) = \log_2 (1+4) + 8 \approx 2.321928094887362 + 8 = 10.321928094887362 \)[/tex]
- For [tex]\( x = 2 \)[/tex], [tex]\( g(2) = \log_2 (2+4) + 8 \approx 2.584962500721156 + 8 = 10.584962500721156 \)[/tex]
- For [tex]\( x = 4 \)[/tex], [tex]\( g(4) = \log_2 (4+4) + 8 = 3.0 + 8 = 11.0 \)[/tex]
- For [tex]\( x = 8 \)[/tex], [tex]\( g(8) = \log_2 (8+4) + 8 \approx 3.584962500721156 + 8 = 11.584962500721156 \)[/tex]
Thus, the final values of [tex]\( g(x) \)[/tex] are [tex]\([10.321928094887362, 10.584962500721156, 11.0, 11.584962500721156] \)[/tex].
### Summary of the transformations and features:
- The function [tex]\( g(x) = \log_2 (x+4) + 8 \)[/tex] is obtained by translating [tex]\( \log_2 x \)[/tex] 4 units to the left and then 8 units up.
- This results in a set of transformations:
- Original function values [tex]\( f(x) \)[/tex] for [tex]\([1, 2, 4, 8] \)[/tex] are [tex]\([0.0, 1.0, 2.0, 3.0] \)[/tex].
- After translating horizontally by 4 units, the values become [tex]\([2.321928094887362, 2.584962500721156, 3.0, 3.584962500721156] \)[/tex].
- After adding 8, the final values of [tex]\( g(x) \)[/tex] are [tex]\([10.321928094887362, 10.584962500721156, 11.0, 11.584962500721156] \)[/tex].
The features of the new function [tex]\( g(x) \)[/tex] include a shifted graph of [tex]\( \log_2 x \)[/tex] both horizontally and vertically.
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