Solved: ?For the following exercises, eliminate the parameter t to rewrite the

What is a system of parametric equations?

Some examples of a third parameter are time, length, speed, and scale. Explain when time is used as a parameter.

Explain how to eliminate a parameter given a set of parametric equations.

What is a benefit of writing a system of parametric equations as a Cartesian equation?

What is a benefit of using parametric equations?

Why are there many sets of parametric equations to represent on Cartesian function?

For the following exercises, eliminate the parameter t to rewrite the parametric equation as a Cartesian equation. \(\left\{\begin{array}{l} x(t)=5-t \\ y(t)=8-2 t \end{array}\right\) Text Transcription: x(t)=5-t _ y(t)=8-2t

For the following exercises, eliminate the parameter t to rewrite the parametric equation as a Cartesian equation. \(\left\{\begin{array}{l} x(t)=6-3 t \\ y(t)=10-t \end{array}\right.\) Text Transcription: x(t)=6-3t _ y(t)=10-t

For the following exercises, eliminate the parameter t to rewrite the parametric equation as a Cartesian equation. \(\left\{\begin{array}{l} x(t)=2 t+1 \\ y(t)=3 \sqrt{t} \end{array}\right.\) Text Transcription: x(t)=2t+1 _ y(t)=3 sqrt t

For the following exercises, eliminate the parameter t to rewrite the parametric equation as a Cartesian equation. \(\left\{\begin{array}{l} x(t)=3 t-1 \\ y(t)=2 t^{2} \end{array}\right.\) Text Transcription: x(t)=3t-1 _ y(t)=2 t^2

For the following exercises, eliminate the parameter t to rewrite the parametric equation as a Cartesian equation. \(\left\{\begin{array}{l} x(t)=2 e^{t} \\ y(t)=1-5 t \end{array}\right.\) Text Transcription: x(t)=2e^t _ y(t)=1-5t

For the following exercises, eliminate the parameter t to rewrite the parametric equation as a Cartesian equation. \(\left\{\begin{array}{l} x(t)=e^{-2 t} \\ y(t)=2 e^{-t} \end{array}\right.\) Text Transcription: x(t)=e^-2t _ y(t)=2e^-t

For the following exercises, eliminate the parameter t to rewrite the parametric equation as a Cartesian equation. \(\left\{\begin{array}{l} x(t)=4 \log (t) \\ y(t)=3+2 t \end{array}\right.\) Text Transcription: x(t)=4 log(t) _ y(t)=3+2t

For the following exercises, eliminate the parameter t to rewrite the parametric equation as a Cartesian equation. \(\left\{\begin{array}{l} x(t)=\log (2 t) \\ y(t)=\sqrt{t-1} \end{array}\right.\) Text Transcription: x(t)=log (2t) _ y(t)=\sqrt t-1

For the following exercises, eliminate the parameter t to rewrite the parametric equation as a Cartesian equation. \(\left\{\begin{array}{l} x(t)=t^{3}-t \\ y(t)=2 t \end{array}\right.\) Text Transcription: x(t)=t^3 -t _ y(t)=2t

For the following exercises, eliminate the parameter t to rewrite the parametric equation as a Cartesian equation. \(\left\{\begin{array}{l} x(t)=t-t^{4} \\ y(t)=t+2 \end{array}\right.\) Text Transcription: x(t)=t-t^4 _ y(t)=t+2

For the following exercises, eliminate the parameter t to rewrite the parametric equation as a Cartesian equation. \(\left\{\begin{array}{l} x(t)=e^{2 t} \\ y(t)=e^{6 t} \end{array}\right.\) Text Transcription: x(t)=e^2t _ y(t)=e^6t

For the following exercises, eliminate the parameter t to rewrite the parametric equation as a Cartesian equation. \(\left\{\begin{array}{l} x(t)=t^{5} \\ y(t)=t^{10} \end{array}\right.\) Text Transcription: x(t)=t^5 _ y(t)=t^10

For the following exercises, eliminate the parameter t to rewrite the parametric equation as a Cartesian equation. \(\left\{\begin{array}{l} x(t)=4 \cos t \\ y(t)=5 \sin t \end{array}\right.\) Text Transcription: x(t)=4 cos t _ y(t)=5 sin t

For the following exercises, eliminate the parameter t to rewrite the parametric equation as a Cartesian equation. \(\left\{\begin{array}{l} x(t)=3 \sin t \\ y(t)=6 \cos t \end{array}\right.\) Text Transcription: x(t)=3 sin t _ y(t)=6 cos t

For the following exercises, eliminate the parameter t to rewrite the parametric equation as a Cartesian equation. \(\left\{\begin{array}{l} x(t)=2 \cos ^{2} t \\ y(t)=-\sin \end{array}\right.\) Text Transcription: x(t)=2 cos ^2 t _ y(t)=-sin t

For the following exercises, eliminate the parameter t to rewrite the parametric equation as a Cartesian equation. \(\left\{\begin{array}{l} x(t)=\cos t+4 \\ y(t)=2 \sin ^{2} t \end{array}\right.\) Text Transcription: x(t)= cos t+4 _ y(t)=2 sin^2 t

For the following exercises, eliminate the parameter t to rewrite the parametric equation as a Cartesian equation. \(\left\{\begin{array}{l} x(t)=t-1 \\ y(t)=t^{2} \end{array}\right.\) Text Transcription: x(t)=t-1 _ y(t)=t^2

For the following exercises, eliminate the parameter t to rewrite the parametric equation as a Cartesian equation. \(\left\{\begin{array}{l} x(t)=-t \\ y(t)=t^{3}+1 \end{array}\right.\) Text Transcription: x(t)=-t _ y(t)=t^3 +1

For the following exercises, eliminate the parameter t to rewrite the parametric equation as a Cartesian equation. \(\left\{\begin{array}{l} x(t)=2 t-1 \\ y(t)=t^{3}-2 \end{array}\right.\) Text Transcription: x(t)=2t-1 _ y(t)=t^3 -2

For the following exercises, rewrite the parametric equation as a Cartesian equation by building an x-y table. \(\left\{\begin{array}{l} x(t)=2 t-1 \\ y(t)=t+4 \end{array}\right.\) Text Transcription: x(t)=2 t-1 _ y(t)=t+4

For the following exercises, rewrite the parametric equation as a Cartesian equation by building an x-y table. \(\left\{\begin{array}{l} x(t)=4-t \\ y(t)=3 t+2 \end{array}\right.\) Text Transcription: x(t)=4-t _ y(t)=3t+2

For the following exercises, rewrite the parametric equation as a Cartesian equation by building an x-y table. \(\left\{\begin{array}{l} x(t)=2 t-1 \\ y(t)=5 t \end{array}\right.\) Text Transcription: x(t)=2t-1 _ y(t)=5 t

For the following exercises, rewrite the parametric equation as a Cartesian equation by building an x-y table. \(\left\{\begin{array}{l} x(t)=4 t-1 \\ y(t)=4 t+2 \end{array}\right.\) Text Transcription: x(t)=4t-1 _ y(t)=4t+2

For the following exercises, parameterize (write parametric equations for) each Cartesian equation by setting x(t) = t or by setting y(t) = t. \(y(x)=3 x^{2}+3\) Text Transcription: y(x)=3x^2 +3

For the following exercises, parameterize (write parametric equations for) each Cartesian equation by setting x(t) = t or by setting y(t) = t. y(x) = 2 sin x + 1

For the following exercises, parameterize (write parametric equations for) each Cartesian equation by setting x(t) = t or by setting y(t) = t. x(y) = 3 log (y) + y

For the following exercises, parameterize (write parametric equations for) each Cartesian equation by setting x(t) = t or by setting y(t) = t. \(x(y)=\sqrt{y}+2 y\) Text Transcription: x(y)= sqrt y +2y

What is a system of parametric equations?

Some examples of a third parameter are time, length, speed, and scale. Explain when time is used as a parameter.

Explain how to eliminate a parameter given a set of parametric equations.

What is a benefit of writing a system of parametric equations as a Cartesian equation?

What is a benefit of using parametric equations?

Why are there many sets of parametric equations to represent on Cartesian function?

For the following exercises, eliminate the parameter t to rewrite the parametric equation as a Cartesian equation. \(\left\{\begin{array}{l} x(t)=5-t \\ y(t)=8-2 t \end{array}\right\) Text Transcription: x(t)=5-t _ y(t)=8-2t

For the following exercises, eliminate the parameter t to rewrite the parametric equation as a Cartesian equation. \(\left\{\begin{array}{l} x(t)=6-3 t \\ y(t)=10-t \end{array}\right.\) Text Transcription: x(t)=6-3t _ y(t)=10-t

For the following exercises, eliminate the parameter t to rewrite the parametric equation as a Cartesian equation. \(\left\{\begin{array}{l} x(t)=2 t+1 \\ y(t)=3 \sqrt{t} \end{array}\right.\) Text Transcription: x(t)=2t+1 _ y(t)=3 sqrt t

For the following exercises, eliminate the parameter t to rewrite the parametric equation as a Cartesian equation. \(\left\{\begin{array}{l} x(t)=3 t-1 \\ y(t)=2 t^{2} \end{array}\right.\) Text Transcription: x(t)=3t-1 _ y(t)=2 t^2

For the following exercises, eliminate the parameter t to rewrite the parametric equation as a Cartesian equation. \(\left\{\begin{array}{l} x(t)=2 e^{t} \\ y(t)=1-5 t \end{array}\right.\) Text Transcription: x(t)=2e^t _ y(t)=1-5t

For the following exercises, eliminate the parameter t to rewrite the parametric equation as a Cartesian equation. \(\left\{\begin{array}{l} x(t)=e^{-2 t} \\ y(t)=2 e^{-t} \end{array}\right.\) Text Transcription: x(t)=e^-2t _ y(t)=2e^-t

For the following exercises, eliminate the parameter t to rewrite the parametric equation as a Cartesian equation. \(\left\{\begin{array}{l} x(t)=4 \log (t) \\ y(t)=3+2 t \end{array}\right.\) Text Transcription: x(t)=4 log(t) _ y(t)=3+2t

For the following exercises, eliminate the parameter t to rewrite the parametric equation as a Cartesian equation. \(\left\{\begin{array}{l} x(t)=\log (2 t) \\ y(t)=\sqrt{t-1} \end{array}\right.\) Text Transcription: x(t)=log (2t) _ y(t)=\sqrt t-1

For the following exercises, eliminate the parameter t to rewrite the parametric equation as a Cartesian equation. \(\left\{\begin{array}{l} x(t)=t^{3}-t \\ y(t)=2 t \end{array}\right.\) Text Transcription: x(t)=t^3 -t _ y(t)=2t

For the following exercises, eliminate the parameter t to rewrite the parametric equation as a Cartesian equation. \(\left\{\begin{array}{l} x(t)=t-t^{4} \\ y(t)=t+2 \end{array}\right.\) Text Transcription: x(t)=t-t^4 _ y(t)=t+2

For the following exercises, eliminate the parameter t to rewrite the parametric equation as a Cartesian equation. \(\left\{\begin{array}{l} x(t)=e^{2 t} \\ y(t)=e^{6 t} \end{array}\right.\) Text Transcription: x(t)=e^2t _ y(t)=e^6t

For the following exercises, eliminate the parameter t to rewrite the parametric equation as a Cartesian equation. \(\left\{\begin{array}{l} x(t)=t^{5} \\ y(t)=t^{10} \end{array}\right.\) Text Transcription: x(t)=t^5 _ y(t)=t^10

For the following exercises, eliminate the parameter t to rewrite the parametric equation as a Cartesian equation. \(\left\{\begin{array}{l} x(t)=4 \cos t \\ y(t)=5 \sin t \end{array}\right.\) Text Transcription: x(t)=4 cos t _ y(t)=5 sin t

For the following exercises, eliminate the parameter t to rewrite the parametric equation as a Cartesian equation. \(\left\{\begin{array}{l} x(t)=3 \sin t \\ y(t)=6 \cos t \end{array}\right.\) Text Transcription: x(t)=3 sin t _ y(t)=6 cos t

For the following exercises, eliminate the parameter t to rewrite the parametric equation as a Cartesian equation. \(\left\{\begin{array}{l} x(t)=2 \cos ^{2} t \\ y(t)=-\sin \end{array}\right.\) Text Transcription: x(t)=2 cos ^2 t _ y(t)=-sin t

For the following exercises, eliminate the parameter t to rewrite the parametric equation as a Cartesian equation. \(\left\{\begin{array}{l} x(t)=\cos t+4 \\ y(t)=2 \sin ^{2} t \end{array}\right.\) Text Transcription: x(t)= cos t+4 _ y(t)=2 sin^2 t

For the following exercises, eliminate the parameter t to rewrite the parametric equation as a Cartesian equation. \(\left\{\begin{array}{l} x(t)=t-1 \\ y(t)=t^{2} \end{array}\right.\) Text Transcription: x(t)=t-1 _ y(t)=t^2

For the following exercises, eliminate the parameter t to rewrite the parametric equation as a Cartesian equation. \(\left\{\begin{array}{l} x(t)=-t \\ y(t)=t^{3}+1 \end{array}\right.\) Text Transcription: x(t)=-t _ y(t)=t^3 +1

For the following exercises, eliminate the parameter t to rewrite the parametric equation as a Cartesian equation. \(\left\{\begin{array}{l} x(t)=2 t-1 \\ y(t)=t^{3}-2 \end{array}\right.\) Text Transcription: x(t)=2t-1 _ y(t)=t^3 -2

For the following exercises, rewrite the parametric equation as a Cartesian equation by building an x-y table. \(\left\{\begin{array}{l} x(t)=2 t-1 \\ y(t)=t+4 \end{array}\right.\) Text Transcription: x(t)=2 t-1 _ y(t)=t+4

For the following exercises, rewrite the parametric equation as a Cartesian equation by building an x-y table. \(\left\{\begin{array}{l} x(t)=4-t \\ y(t)=3 t+2 \end{array}\right.\) Text Transcription: x(t)=4-t _ y(t)=3t+2

For the following exercises, rewrite the parametric equation as a Cartesian equation by building an x-y table. \(\left\{\begin{array}{l} x(t)=2 t-1 \\ y(t)=5 t \end{array}\right.\) Text Transcription: x(t)=2t-1 _ y(t)=5 t

For the following exercises, rewrite the parametric equation as a Cartesian equation by building an x-y table. \(\left\{\begin{array}{l} x(t)=4 t-1 \\ y(t)=4 t+2 \end{array}\right.\) Text Transcription: x(t)=4t-1 _ y(t)=4t+2

For the following exercises, parameterize (write parametric equations for) each Cartesian equation by setting x(t) = t or by setting y(t) = t. \(y(x)=3 x^{2}+3\) Text Transcription: y(x)=3x^2 +3

For the following exercises, parameterize (write parametric equations for) each Cartesian equation by setting x(t) = t or by setting y(t) = t. y(x) = 2 sin x + 1

For the following exercises, parameterize (write parametric equations for) each Cartesian equation by setting x(t) = t or by setting y(t) = t. x(y) = 3 log (y) + y

For the following exercises, parameterize (write parametric equations for) each Cartesian equation by setting x(t) = t or by setting y(t) = t. \(x(y)=\sqrt{y}+2 y\) Text Transcription: x(y)= sqrt y +2y

For the following exercises, parameterize (write parametric equations for) each Cartesian equation by using x(t) = a cos t and y(t) = b sin t. Identify the curve. \(\frac{x^{2}}{4}+\frac{y^{2}}{9}=1\) Text Transcription: frac x^2 /4 +frac y^2 /9 =1

For the following exercises, parameterize (write parametric equations for) each Cartesian equation by using x(t) = a cos t and y(t) = b sin t. Identify the curve. \(\frac{x^{2}}{16}+\frac{y^{2}}{36}=1\) Text Transcription: frac x^2 /16 +frac y^2 /36 =1

For the following exercises, parameterize (write parametric equations for) each Cartesian equation by using x(t) = a cos t and y(t) = b sin t. Identify the curve. \(x^{2}+y^{2}=16\) Text Transcription: x^2 +y^2 =16

For the following exercises, parameterize (write parametric equations for) each Cartesian equation by using x(t) = a cos t and y(t) = b sin t. Identify the curve. \(x^{2}+y^{2}=10\) Text Transcription: x^2 +y^2 =10

For the following exercises, use the table feature in the graphing calculator to determine whether the graphs intersect. \(\left\{\begin{array}{l} x_{1}(t)=3 t \\ y_{1}(t)=2 t-1 \end{array}\right.\) and \(\left\{\begin{array}{l} x_{2}(t)=t+3 \\ y_{2}(t)=4 t-4 \end{array}\right.\) Text Transcription: x_1 (t)=3t _ y_1 (t)=2t-1 and x_2 (t)=t+3 _ y_2 (t)=4t-4

For the following exercises, use the table feature in the graphing calculator to determine whether the graphs intersect. \(\left\{\begin{array}{l} x_{1}(t)=t^{2} \\ y_{1}(t)=2 t-1 \end{array}\right.\) and \(\left\{\begin{array}{l} x_{2}(t)=-t+6 \\ y_{2}(t)=t+1 \end{array}\right.\) Text Transcription: x_1 (t)=t^2 _ y_1 (t)=2t-1 and x_2 (t)=-t+6 _ y_2 (t)=t+1

For the following exercises, use a graphing calculator to complete the table of values for each set of parametric equations. \(\left\{\begin{array}{l} x_{1}(t)=3 t^{2}-3 t+7 \\ y_{1}(t)=2 t+3 \end{array}\right.\) Text Transcription: x_1 (t)=3t^2 -3t+7 _ y_1 (t)=2t+3

For the following exercises, use a graphing calculator to complete the table of values for each set of parametric equations. \(\left\{\begin{array}{l} x_{1}(t)=t^{2}-4 \\ y_{1}(t)=2 t^{2}-1 \end{array}\right.\) Text Transcription: x_1 (t)=t^2 -4 _ y_1 (t)=2 t^2 -1

For the following exercises, use a graphing calculator to complete the table of values for each set of parametric equations. \(\left\{\begin{array}{l} x_{1}(t)=t^{4} \\ y_{1}(t)=t^{3}+4 \end{array}\right.\) Text Transcription: x_1 (t)=t^4 _ y_1 (t)=t^3 +4

Find two different sets of parametric equations for \(y=(x+1)^{2}\). Text Transcription: y=(x+1)^2

Find two different sets of parametric equations for \(y=x^{2}-4 x+4\). Text Transcription: y=x^2 -4x+4

Parameterize the line from (3, 0) to (?2, ?5) so that the line is at (3, 0) at t = 0, and at (?2, ?5) at t = 1.

Parameterize the line from (?1, 0) to (3, ?2) so that the line is at (?1, 0) at t = 0, and at (3, ?2) at t = 1.

Parameterize the line from (?1, 5) to (2, 3) so that the line is at (?1, 5) at t = 0, and at (2, 3) at t = 1.

Parameterize the line from (4, 1) to (6, ?2) so that the line is at (4, 1) at t = 0, and at (6, ?2) at t = 1.

Find two different sets of parametric equations for y = 3x ? 2.