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Identify the parametric equations that represent the same graph as the following parametric equations. Select all that apply.x=4ty=t^2-1

Identify The Parametric Equations That Represent The Same Graph As The Following Parametric Equations Select All That Applyx4tyt21 class=

Sagot :

Given the equations:

[tex]\begin{gathered} x=4t \\ \\ y=t^2-1 \end{gathered}[/tex]

Let's identify the parametric equations which represent the same graph as the given equations.

Here, we have:

Rewrite the first equation for t

[tex]t=\frac{x}{4}[/tex]

Now, plug in x/4 for t in the second equation:

[tex]\begin{gathered} y=(\frac{x}{4})^2-1 \\ \\ y=\frac{x^2}{4^2}-1 \\ \\ y=\frac{x^2}{16}-1 \end{gathered}[/tex]

Also we have:

[tex]t^2=\frac{x^2}{16}[/tex]

Rewrite equation 2 for t²

[tex]t^2=y+1[/tex]

Hence, we have:

[tex]\begin{gathered} \frac{x^2}{16}=t^2=y+1 \\ \\ \frac{x^2}{16}=y+1 \\ \\ y=\frac{x^2}{16}-1 \end{gathered}[/tex]

Now, let's solve for the following to confirm:

[tex]\begin{gathered} \text{ When x = 3t} \\ Substitute\text{ 3t for x:} \\ y=\frac{(3t)^2}{16}-1 \\ \\ y=\frac{9t^2}{16}-1 \\ \end{gathered}[/tex]

Option B:

[tex]\begin{gathered} \text{ When x = 2t:} \\ \text{ Substitute 2t for x} \\ y=\frac{(2t)^2}{16}-1 \\ \\ y=\frac{4t^2}{16}-1 \\ \\ y=\frac{t^2}{4}-1 \\ \\ Option\text{ B is correct.} \end{gathered}[/tex]

Option C:

[tex]\begin{gathered} \text{ When x =8t} \\ \text{ Substitute 8t for x:} \\ y=\frac{(8t)^2}{16}-1 \\ \\ y=\frac{64t^2}{16}-1 \\ \\ y=4t^2-1 \\ \\ Option\text{ C is correct.} \end{gathered}[/tex]

Therefore, the correct parametric equations are:

[tex]\begin{gathered} B.\text{ x = 2t} \\ y=\frac{t^2}{4}-1 \\ \\ \\ \\ C.x=8t \\ y=4t^2-1 \end{gathered}[/tex]

ANSWER:

• B. x = 2t

y = t²/4 - 1

• C. x = 8t

y = 4t² - 1