To determine which parametric equations represent [tex]\( y = -|x| \)[/tex], we need to examine each pair of parametric equations given and verify if they satisfy this relationship. Let’s analyze each one step by step.
1. Equation Pair: [tex]\( x = -t \)[/tex] and [tex]\( y = | -t | \)[/tex]
- Substitute [tex]\( x = -t \)[/tex] into [tex]\( y = | -t | \)[/tex]:
[tex]\[
y = | -t | = | x |
\][/tex]
- Since [tex]\( y = | x | \)[/tex], this does not satisfy [tex]\( y = -| x | \)[/tex]. Thus, this pair does not represent [tex]\( y = -|x| \)[/tex].
2. Equation Pair: [tex]\( x = t^2 \)[/tex] and [tex]\( y = -| t^2 | \)[/tex]
- Substitute [tex]\( x = t^2 \)[/tex] into [tex]\( y = -| t^2 | \)[/tex]:
[tex]\[
y = -| t^2 | = -| x | \text{ (since \( t^2 \geq 0 \) for all real \( t \))}
\][/tex]
- This satisfies [tex]\( y = -|x| \)[/tex]. Thus, this pair represents [tex]\( y = -|x| \)[/tex].
3. Equation Pair: [tex]\( x = 2t \)[/tex] and [tex]\( y = -| 2t | \)[/tex]
- Substitute [tex]\( x = 2t \)[/tex] into [tex]\( y = -| 2t | \)[/tex]:
[tex]\[
y = -| 2t | = -| 2x | = -2|x|
\][/tex]
- However, [tex]\( y = -| 2x | \neq -| x | \)[/tex]. This does not satisfy [tex]\( y = -|x| \)[/tex]. Thus, this pair does not represent [tex]\( y = -|x| \)[/tex].
4. Equation Pair: [tex]\( x = t^3 \)[/tex] and [tex]\( y = | t^3 | \)[/tex]
- Substitute [tex]\( x = t^3 \)[/tex] into [tex]\( y = | t^3 | \)[/tex]:
[tex]\[
y = | t^3 | = | x |
\][/tex]
- Since [tex]\( y = | x | \)[/tex], this does not satisfy [tex]\( y = -| x | \)[/tex]. Thus, this pair does not represent [tex]\( y = -|x| \)[/tex].
5. Equation Pair: [tex]\( x = t + 4 \)[/tex] and [tex]\( y = -| t + 4 | \)[/tex]
- Substitute [tex]\( x = t + 4 \)[/tex] into [tex]\( y = -| t + 4 | \)[/tex]:
[tex]\[
y = -| t + 4 | = -| x |
\][/tex]
- This satisfies [tex]\( y = -|x| \)[/tex]. Thus, this pair represents [tex]\( y = -|x| \)[/tex].
In conclusion, the parametric equations that represent [tex]\( y = -| x | \)[/tex] are:
- [tex]\( x = t^2 \)[/tex] and [tex]\( y = -| t^2 | \)[/tex]
- [tex]\( x = t + 4 \)[/tex] and [tex]\( y = -| t + 4 | \)[/tex]
