Ashleenbondidn
Answered

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

- Shifts [tex]\( f(x) \)[/tex] 4 units right
- Stretches [tex]\( f(x) \)[/tex] by a factor of 4 away from the [tex]\( x \)[/tex]-axis
- Compresses [tex]\( f(x) \)[/tex] by a factor of [tex]\( \frac{1}{4} \)[/tex] toward the [tex]\( y \)[/tex]-axis
- Shifts [tex]\( f(x) \)[/tex] 4 units down

Sagot :

GinnyAnsweridn
To match each transformation of [tex]\( f(x) = 2x - 6 \)[/tex] with its description, follow these steps:

1. Identify the shifts and transformations based on the given function and transformations:

- When we write the function in terms of a transformation, we consider alterations in the form [tex]\( g(x) = a \cdot f(b \cdot (x - h)) + k \)[/tex] where [tex]\( a \)[/tex] causes a vertical stretch or compression, [tex]\( b \)[/tex] affects horizontal stretch or compression, [tex]\( h \)[/tex] causes a horizontal shift, and [tex]\( k \)[/tex] causes a vertical shift.

2. Examine each given transformation:

- [tex]\( g(x) = 2x - 2 \)[/tex]: Compare with [tex]\( f(x) \)[/tex]:
[tex]\( 2x - 2 = 2(x - (-2)) = 2(x + 4) \)[/tex]
Hence, this transformation is [tex]\( f(x) \)[/tex] shifted 4 units to the right.

- [tex]\( g(x) = 8x - 24 \)[/tex]: Compare with [tex]\( f(x) \)[/tex]:
[tex]\( 8x - 24 = 4 \cdot (2x - 6) = 4 \cdot f(x) \)[/tex]
Hence, this transformation stretches [tex]\( f(x) \)[/tex] by a factor of 4 away from the x-axis.

- [tex]\( g(x) = 8x - 6 \)[/tex]: Compare with [tex]\( f(x) \)[/tex]:
This does not match any transformation needed so discard this.

- [tex]\( g(x) = 2x - 14 \)[/tex]: Compare with [tex]\( f(x) \)[/tex]:
This does not match any transformation needed so discard this.

- [tex]\( g(x) = 2x - 10 \)[/tex]: Compare with [tex]\( f(x) \)[/tex]:
[tex]\( 2x - 10 = 2x - (6 + 4) = f(x) - 4 \)[/tex]
Hence, this transformation shifts [tex]\( f(x) \)[/tex] 4 units down.

- [tex]\( g(x) = 8x - 4 \)[/tex]: Compare with [tex]\( f(x) \)[/tex]:
This does not match any transformation needed so discard this.

3. Match the appropriate transformations to descriptions:

- [tex]\( g(x)=2x - 2 \)[/tex]: shifts [tex]\( f(x) \)[/tex] 4 units right
- [tex]\( g(x)=8x - 24 \)[/tex]: stretches [tex]\( f(x) \)[/tex] by a factor of 4 away from the [tex]\( x \)[/tex]-axis
- [tex]\( g(x)=8x - 6 \)[/tex]: compresses [tex]\( f(x) \)[/tex] by a factor of [tex]\( \frac{1}{4} \)[/tex] toward the [tex]\( y \)[/tex]-axis
- [tex]\( g(x)=2x - 10 \)[/tex]: shifts [tex]\( f(x) 4 \)[/tex] units down

So, the completed pairs should be:

- [tex]\( g(x) = 2x - 2 \)[/tex] [tex]$\quad \implies \quad$[/tex] shifts [tex]\( f(x) \)[/tex] 4 units right
- [tex]\( g(x) = 8x - 24 \)[/tex] [tex]$\quad \implies \quad$[/tex] stretches [tex]\( f(x) \)[/tex] by a factor of 4 away from the [tex]\( x \)[/tex]-axis
- [tex]\( g(x) = 8x - 6 \)[/tex] [tex]$\quad \implies \quad$[/tex] compresses [tex]\( f(x) \)[/tex] by a factor of [tex]\( \frac{1}{4} \)[/tex] toward the [tex]\( y \)[/tex]-axis
- [tex]\( g(x) = 2x - 10 \)[/tex] [tex]$\quad \implies \quad$[/tex] shifts [tex]\( f(x) 4 \)[/tex] units down
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