Alright, let's take each transformation of [tex]\( f(x) = 2x - 6 \)[/tex] one by one and match it with the descriptions provided:
1. Compresses [tex]\( f(x) \)[/tex] by a factor of [tex]\( \frac{1}{4} \)[/tex] toward the [tex]\( y \)[/tex]-axis:
Compressing [tex]\( f(x) = 2x - 6 \)[/tex] by a factor of [tex]\( \frac{1}{4} \)[/tex] toward the [tex]\( y \)[/tex]-axis means we multiply the coefficient of [tex]\( x \)[/tex] by 4.
[tex]\[
f\left(\frac{x}{4}\right) = 2\left(\frac{x}{4}\right) - 6 = \frac{1}{2}x - 6 \quad (\text{This transformation is not obtained directly from the provided functions, hence discard it})
\][/tex]
Alternatively, compress the entire function:
[tex]\( g(x) = \frac{f(x)}{4} = \frac{2x-6}{4} = \frac{1}{2}x - 1.5 (\text{none match, another discard}) \)[/tex]
The correct function provided for this case would be:
[tex]\( g(x) = 8x - 24 \)[/tex]
2. Stretches [tex]\( f(x) \)[/tex] by a factor of 4 away from the [tex]\( x \)[/tex]-axis:
Stretching [tex]\( f(x) = 2x - 6 \)[/tex] by a factor of 4 means we multiply the entire function by 4:
[tex]\[
g(x) = 4(2x - 6) = 8x - 24
\][/tex]
The function [tex]\( g(x) = 8x - 24 \)[/tex] matches this transformation.
3. Shifts [tex]\( f(x) \)[/tex] 4 units to the right:
Shifting the function to the right by 4 units means we replace [tex]\( x \)[/tex] by [tex]\( x - 4 \)[/tex]:
[tex]\[
f(x - 4) = 2(x - 4) - 6 = 2x - 8 - 6 = 2x - 14
\][/tex]
The function [tex]\( g(x) = 2x - 14 \)[/tex] matches this transformation.
4. Shifts [tex]\( f(x) \)[/tex] 4 units down:
Shifting the function downward by 4 units means we subtract 4 from the whole function:
[tex]\[
f(x) - 4 = (2x - 6) - 4 = 2x - 10
\][/tex]
The function [tex]\( g(x) = 2x - 10 \)[/tex] matches this transformation.
By matching the transformations and descriptions:
- Compresses [tex]\( f(x) \)[/tex] by a factor of [tex]\( \frac{1}{4} \)[/tex] toward the [tex]\( y \)[/tex]-axis: [tex]\( \(g(x) = 8x-24\)[/tex]
- Stretches [tex]\( f(x) \)[/tex] by a factor of 4 away from the [tex]\( x \)[/tex]-axis: [tex]\( \(g(x) = 8x-24\)[/tex]
- Shifts [tex]\( f(x) 4 \)[/tex] units right: [tex]\( \(g(x) = 2x-14\)[/tex]
- Shifts [tex]\( f(x) 4 \)[/tex] units down: [tex]\( \(g(x) = 2x-10\)[/tex]
