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Which parametric equations represent [tex]$y=-|x|$[/tex]? Check all that apply.

A. [tex]x=-t[/tex] and [tex]y=|-t|[/tex]

B. [tex]x=t^2[/tex] and [tex]y=-\left|t^2\right|[/tex]

C. [tex]x=2t[/tex] and [tex]y=-|2t|[/tex]

D. [tex]x=t^3[/tex] and [tex]y=\left|t^3\right|[/tex]

E. [tex]x=t+4[/tex] and [tex]y=-|t+4|[/tex]

Sagot :

GinnyAnsweridn
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]
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