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Drag the tiles to the correct boxes to complete the pairs. Not all tiles will be used.

Consider the function.

[tex]\[ f(x) = 2x - 6 \][/tex]

Match each transformation of [tex]\( f(x) \)[/tex] with its description.

[tex]\[ g(x) = 2x - 2 \][/tex]
[tex]\[ \begin{array}{l}
g(x) = 8x - 6 \\
g(x) = 8x - 24 \\
g(x) = 8x - 4 \\
g(x) = 2x - 14 \\
g(x) = 2x - 10 \\
\end{array} \][/tex]

[tex]\[
\begin{array}{l}
\text{compresses } f(x) \text{ by a factor of } \frac{1}{4} \text{ toward the } y\text{-axis} \quad \square \\
\text{stretches } f(x) \text{ by a factor of 4 away from the } x\text{-axis} \quad \square \\
\text{shifts } f(x) \text{ 4 units right} \quad \square \\
\text{shifts } f(x) \text{ 4 units down} \quad \square \\
\end{array}
\][/tex]

Sagot :

GinnyAnsweridn
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]
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