Question

eliminate the parameter t to find a cartesian equation in the form x = f(y) for: \

$$\begin{cases} x(t) = -4t^2 \\\\ y(t) = 9 + 1t \\end{cases}$$

the resulting equation can be written as x = \boxed{} question help: \boxed{video}

Explanation:

Step1: Solve for \( t \) from \( y(t) \)

Given \( y = 9 + t \) (assuming the typo is corrected, original \( y(t)=9 + 1t \)), we solve for \( t \):
\( t = y - 9 \)

Step2: Substitute \( t \) into \( x(t) \)

We have \( x = -4t^2 \). Substitute \( t = y - 9 \) into this equation:
\( x = -4(y - 9)^2 \)
Expand \( (y - 9)^2 \):
\( (y - 9)^2 = y^2 - 18y + 81 \)
So \( x = -4(y^2 - 18y + 81) = -4y^2 + 72y - 324 \)

Answer:

\( -4y^2 + 72y - 324 \) (or in factored form \( -4(y - 9)^2 \))